Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

We show that for any lattice Veech group in the mapping class group $\mathrm{Mod}(S)$ of a closed surface $S$, the associated $\pi_1 S$--extension group is a hierarchically hyperbolic group. As a consequence, we prove that any such extension group is quasi-isometrically rigid.


Introduction
This paper studies geometric properties of surface group extensions and how these relate to their defining subgroups of mapping class groups. Let S be a closed, connected, oriented surface of genus at least 2. Recall that a π 1 S-extension of a group G is a short exact sequence of the form 1 Ñ π 1 S Ñ Γ Ñ G Ñ 1.
Such extensions are in bijective correspondence with monodromy homomorphisms from G to the extended mapping class group Mod˘pSq -Outpπ 1 Sq of the surface. Alternatively, these groups Γ are precisely the fundamental groups of S-bundles.
Many advances in the study of mapping class groups have been motivated by a longstanding but incomplete analogy between hyperbolic space H n and the Teichmüller space T pSq of a surface. In the theory of Kleinian groups, a discrete group of isometries of H n is convex cocompact if it acts cocompactly on an invariant, convex subset. Farb and Mosher [FM02a] adapted this notion to mapping class groups by defining a subgroup G ď Mod˘pSq to be convex cocompact if it acts cocompactly on a quasi-convex subset of T pSq. This has proven to be a fruitful concept with many interesting connections to, for example, the intrinsic geometry of the mapping class group [DT15,BBKL20], and its actions on the curve complex and the boundary of Teichmüller space [KL08a]. Most importantly, the work of Farb-Mosher [FM02a] and Hamenstädt [Ham] remarkably shows that an extension Γ as above is word hyperbolic if and only if the associated monodromy G Ñ Mod˘pSq has finite kernel and convex cocompact image (see also [MS12]).
For Kleinian groups, convex cocompactness is a special case of a more prevalent phenomenon called geometric finiteness, which roughly amounts to acting cocompactly on a convex subset minus horoballs invariant by parabolic subgroups. In [Mos06], Mosher suggested this notion should have an analogous framework in mapping class groups that would extend the geometric connection with surface bundles to a larger class of examples. The prototypical candidates for geometric finiteness are the lattice Veech subgroups; these are special punctured-surface subgroups of ModpSq that arise naturally in the context of Teichmüller dynamics and whose corresponding S-bundles are amenable to study via techniques from flat geometry.
Our prequel paper [DDLS21] initiated an analysis of the π 1 S-extensions associated to lattice Veech subgroups, with the main result being that each such extension Γ admits an action on a hyperbolic spaceÊ that captures much of the geometry of Γ. Building on that work, the first main result of this paper is the following, which provides a concrete answer to [Mos06, Problem 6.2] for lattice Veech groups.
Theorem 1.1. For any lattice Veech subgroup G ă ModpSq, the associated π 1 Sextension group Γ of G is a hierarchically hyperbolic group.
Hierarchical hyperbolicity means that in fact all the geometry of Γ is robustly encoded by hyperbolic spaces. This is exactly the sort of relaxed hyperbolicity for π 1 S-extensions that one hopes should follow from a good definition of geometric finiteness in ModpSq. Thus Theorem 1.1 suggests a possible general theory of geometric finiteness, which we expound upon in §1.4 below.
Hierarchical hyperbolicity has many strong consequences, some of which are detailed in §1.1 below. It also enables, via tools from [BHS21], the proof of our second main result, which answers [Mos06, Problem 5.4]: Theorem 1.2. For any lattice Veech group G ă ModpSq, the associated π 1 Sextension group Γ of G is quasi-isometrically rigid.
The rest of this introduction gives a more in-depth treatment of these results while elaborating on the concepts of, and connections between, hierarchical hyperbolicity, extensions of Veech groups, quasi-isometric rigidity, and geometric finiteness.
1.1. Hierarchical hyperbolicity. The notion of hierarchical hyperbolicity was defined by Behrstock, Hagen, and Sisto [BHS17b] and motivated by the seminal work of Masur and Minsky [MM00]. In short, it provides a framework and toolkit for understanding the coarse geometry of a space/group in terms of interrelated hyperbolic pieces. More precisely, a hierarchically hyperbolic space (HHS) structure on a metric space X is a collection of hyperbolic spaces tCpW qu W PS , arranged in a hierarchical fashion, in which any pair are nested Ď, orthogonal K, or transverse &, along with Lipschitz projections to and between these spaces that together capture the coarse geometry of X. A hierarchically hyperbolic group (HHG) is then an HHS structure on a group that is equivariant with respect to an appropriate action on the union of hyperbolic spaces CpW q. See §4 for details or [BHS17b, BHS19,Sis19] for many examples and further discussion.
(2) Γ is acylindrically hyperbolic and, moreover, its action on the Ď-maximal hyperbolic space in the hierarchy is a universal acylindrical action [ABD21].
As discussed in §1.2 below, further information about Γ can be gleaned from the specific HHG structure constructed in proving Theorem 1.1. We note that the Ď-maximal hyperbolic space of this structure, and thus the universal acylindrical action indicated in Corollary 1.3(2), is simply the spaceÊ from [DDLS21].
1.2. The HHG structure on Γ. In order to describe the HHG structure more precisely and explain its connection to quasi-isometric rigidity in Theorem 1.2, we must first recall some of the structure of Veech groups and their extensions. Let G ă ModpSq be a lattice Veech group and Γ " Γ G the associated extension group. First note that (up to finite index) Γ is naturally the fundamental group of an S-bundleĒ{Γ over a compact surface with boundary (see §2 for details and notation). Each boundary component ofĒ{Γ is virtually the mapping torus of a multi-twist on S, and is thus a graph manifold: the tori in the JSJ decomposition are suspensions of the multi-twist curves.
Graph manifolds admit HHS structures [BHS19] where the maximal hyperbolic space is the Bass-Serre tree dual to the JSJ decomposition, and all other hyperbolic spaces are either quasi-lines or quasi-trees (obtained by coning off the boundaries of the universal covers of the base orbifolds of the Seifert pieces). The stabilizers of the vertices of the Bass-Serre trees are called vertex subgroups, and are precisely the fundamental groups of the Seifert pieces of the JSJ decomposition. We let V denote the disjoint union of the vertices of all Bass-Serre trees associated to the boundary components of the universal coverĒ of this S-bundle. Given v, w P V, we say that these vertices are adjacent if they are connected by an edge in the same Bass-Serre tree.
The HHG structure on the extension group Γ may now be described as follows: Theorem 1.4. Suppose G ă ModpSq is a lattice Veech group with extension group Γ and let Υ 1 , . . . , Υ k ă Γ be representatives of the conjugacy classes of vertex subgroups. Then Γ admits an HHG structure with the following set of hyperbolic spaces and relations among them (ignoring those of diameter ď 4): (1) The maximal hyperbolic spaceÊ is quasi-isometric to the Cayley graph of Γ coned off along the cosets of Υ 1 , . . . , Υ k [DDLS21].
(2) There is a quasi-tree v qt and a quasi-line v ql , for each v P V, and: (a) For all v P V, v qt Kv ql .
(b) For all v, w P V, if v and w are adjacent, then w ql Kv ql and w ql Ď v qt . (c) All other pairs are transverse.
acylindrical action on a hyperbolic space is one in which every generalized loxodromic acts loxodromically [ABD21]. It is shown in [Sis16] that a generalized loxodromic element g of a finitely generated group is necessarily Morse, meaning that in any finite-valence Cayley graph for the group, any pK, Cq-quasi-geodesic with endpoints in the cyclic subgroup xgy stays within controlled distance M " M pK, Cq of xgy. While being Morse is, in general, strictly weaker than being generalized loxodromic, these conditions are in fact equivalent in HHGs [ABD21,Theorem B].
In the case of our extension group Γ, it follows from Corollary 1.3(2) that the generalized loxodromics and Morse elements are precisely those elements acting loxodromically onÊ. In [DDLS21, Theorem 1.1] we characterized these elements in terms of the vertex subgroups of Γ, thus yielding the following: Corollary 1.6. Let Γ be a lattice Veech group extension with vertex subgroups Υ 1 , . . . , Υ k as in Theorem 1.4. The following are equivalent for an infinite order element γ P Γ: ‚ γ is not conjugate into any of the vertex subgroups Υ i ‚ γ is a generalized loxodromic element of Γ ‚ γ is a Morse element of Γ.
1.3. Quasi-isometric rigidity. To state our rigidity theorem, first recall that Γ is (up to finite index) the fundamental group of an S-bundle s E{Γ over a compact surface with boundary. HereĒ is a Γ-invariant truncation of the universalS-bundle over the Teichmüller disk stabilized by the Veech group G. In particular,Ē is quasiisometric to Γ. Let Isomp s Eq and QIp s Eq denote the isometry and quasi-isometry groups of s E, respectively, and let Isom fib p s Eq ď Isomp s Eq denote the subgroup of isometries that map fibers to fibers.
Theorem 1.7. There is an allowable truncationĒ of E such that the natural homomorphisms Isom fib pĒq Ñ IsompĒq Ñ QIpĒq -QIpΓq are all isomorphisms, and Γ ď IsompĒq -QIpΓq has finite index. This is an analog, and indeed was motivated by, Farb and Mosher's [FM02b] theorem that in the case of a surface group extension Γ H associated to a Schottky subgroup H of ModpSq, the natural homomorphism Γ H Ñ QIpΓ H q is injective with finite cokernel. This rigidity also leads to the following strong algebraic consequence: Corollary 1.8. If H is any finitely generated group quasi-isometric to Γ, then H and Γ are weakly commensurable.
In the statement, recall that two groups H 1 , H 2 are weakly commensurable if there are finite normal subgroups N i H i so that the quotients H i {N i have a pair of finite-index subgroups that are isomorphic to each other.
1.4. Motivation and Geometric Finiteness. Before outlining the paper and providing some ideas about the proofs, we provide some speculative discussion. For Kleinian groups-that is, discrete groups of isometries of hyperbolic 3-space-the notion of geometric finiteness is important in the deformation theory of hyperbolic 3-manifolds by the work of Ahlfors [Ahl66] and Greenberg [Gre66]. While the definition has many formulations (see [Mar74,Mas70,Thu86,Bow93]), roughly speaking a group is geometrically finite if it acts cocompactly on a convex subset of hyperbolic 3-space minus a collection of horoballs that are invariant by parabolic subgroups. When there are no parabolic subgroups, geometric finiteness reduces to convex cocompactness: a cocompact action on a convex subset of hyperbolic 3-space.
While there is no deformation theory for subgroups of mapping class groups, Farb and Mosher [FM02a] introduced a notion of convex cocompactness for G ă ModpSq in terms of the action on Teichmüller space T pSq. Their definition requires that G acts cocompactly on a quasi-convex subset for the Teichmüller metric, while Kent and Leininger later proved a variety of equivalent formulations analogous to the Kleinian setting [KL07,KL08a,KL08b]. Farb and Mosher proved that convex cocompactness is equivalent to hyperbolicity of the associated extension group Γ G (with monodromy given by inclusion) when G is virtually free. This equivalence was later proven in general by Hamenstädt [Ham] (see also Mj-Sardar [MS12]), though at the moment the only known examples are virtually free.
The coarse nature of Farb and Mosher's formulation reflects the fact that the Teichmüller metric is far less well-behaved than that of hyperbolic 3-space. Quasiconvexity in the definition is meant to help with the lack of nice local behavior of the Teichmüller metric. It also helps with the global lack of Gromov hyperbolicity (see Masur-Wolf [MW95]), as cocompactness of the action ensures that the quasiconvex subset in the definition is Gromov hyperbolic (see , Minsky [Min96b], and Rafi [Raf14]).
The inclusion of reducible/parabolic mapping classes in a subgroup G ă ModpSq brings the thin parts of T pSq into consideration; these subspaces contain higher rank quasi-flats and even exhibit aspects of positive curvature (see Minsky [Min96a]). This is a main reason why extending the notion of convex cocompactness to geometric finiteness is complicated. These complications are somewhat mitigated in the case of lattice Veech groups. Such subgroups are stabilizers of isometrically and totally geodesically embedded hyperbolic planes, called Teichmüller disks, that have finite area quotients. Thus, the intrinsic hyperbolic geometry agrees with the extrinsic Teichmüller geometry, and as a group of isometries of the hyperbolic plane, a lattice Veech group is geometrically finite. This is why these subgroups serve as a test case for geometric finiteness in the mapping class group. This is also why a subgroup of a Veech group is convex cocompact in ModpSq if and only if it is convex cocompact as a group of isometries of the hyperbolic plane (which also happens if and only if it is finitely generated and contains no parabolic elements).
The action of ModpSq on the curve graph, which is Gromov hyperbolic by work of Masur-Minsky [MM99], provides an additional model for these considerations. Specifically, convex cocompactness is equivalent to the orbit map to the curve graph CpSq being a quasi-isometric embedding with respect to the word metric from a finite generating set (see  and Hamenstädt [Ham]). Viewing geometric finiteness as a kind of "relative convex cocompactness" for Kleinian groups suggests an interesting connection with the curve complex formulation. The connection is best illustrated by the following theorem of Tang [Tan19]. Theorem 1.9 (Tang). For any lattice Veech group G ă ModpSq stabilizing a Teichmüller disk D Ă T pSq, there is a G-equivariant quasi-isometric embedding D el Ñ CpSq, where D el is the path metric space obtained from D by coning off the G-invariant family of horoballs in which D ventures into the thin parts of T pSq.
Farb [Far98] showed that non-cocompact lattices in the group of isometries of hyperbolic space are relatively hyperbolic relative to the parabolic subgroups. For Veech groups, the space D el is quasi-isometric to the (hyperbolic) coned off Cayley graph, illustrating (part of) the relative hyperbolicity of G. We thus propose a kind of "qualified" notion of geometric finiteness with this in mind: Definition 1.10 (Parabolic geometric finiteness). A finitely generated subgroup G ă ModpSq is parabolically geometrically finite if G is relatively hyperbolic, relative to a (possibly trivial) collection of subgroups H " tH 1 , . . . , H k u, and (1) H i contains a finite index, abelian subgroup consisting entirely of multitwists, for each 1 ď i ď k; and (2) the coned off Cayley graph G-equivariantly and quasi-isometrically embeds into CpSq.
When H " ttiduu, we note that the condition is equivalent to G being convex cocompact. By Theorem 1.9, lattice Veech groups are parabolically geometrically finite. In fact, Tang's result is more general and implies that any finitely generated Veech group satisfies this definition. These examples are all virtually free, but other examples include the combination subgroups of Leininger-Reid [LR06], which are isomorphic to fundamental groups of closed surfaces of higher genus, and free products of higher rank abelian groups constructed by Loa [Loa21].
In view of Theorem 1.1, one might formulate the following.
Conjecture 1.11. Let G ă ModpSq be parabolically geometrically finite. Then the π 1 S-extension group Γ of G is a hierarchically hyperbolic group.
We view Definition 1.10 as only a qualified formulation because there are many subgroups of ModpSq that are not relatively hyperbolic but are nevertheless candidates for being geometrically finite in some sense. It is possible that there are different types of geometric finiteness for subgroups of mapping class groups, with Definition 1.10 being among the most restrictive. Other notions might include an HHS structure on the subgroup which is compatible with the ambient one on ModpSq (e.g., hierarchical quasiconvexity [BHS19]). From this perspective, some candidate subgroups that may be considered geometrically finite include: ‚ the whole group ModpSq; ‚ multi-curve stabilizers; ‚ the right-angled Artin subgroups of mapping class groups constructed in [CLM12,Kob12,Run20]; ‚ free and amalgamated products of other examples.
Question 1.12. For each example group G ď ModpSq above, is the associated extension Γ G a hierarchically hyperbolic group?
We note that the answer is 'yes' for the first example, since the extension group is the mapping class group of the surface S with a puncture. Moreover, since our work on this subject first appeared, Russell [Rus21] addressed the second example by proving extensions of multicurve stabilizers are hierarchically hyperbolic groups.
1.5. Outline and proofs. Let us briefly outline the paper and comment on the main structure of the proofs. In §2 we review necessary background material and introduce the objects and notation that will be used throughout the paper. In particular, we define the spaces E andĒ, the latter being a quasi-isometric model for the Veech group extension Γ, as well as the hyperbolic collapsed spaceÊ. All of these were constructed in [DDLS21].
In § §3-4 we prove that the extension group Γ is hierarchically hyperbolic by utilizing a combinatorial criterion from [BHMS20]. Besides hyperbolicity ofÊ, the other hard part of the criterion is an analogue of Bowditch's fineness condition from the context of relative hyperbolicity. Its geometric interpretation is roughly that two cosets of vertex subgroups as above have bounded coarse intersection, aside from the "obvious" exception when the cosets correspond to vertices of the same Bass-Serre tree within distance 2 of each other. To this end, in §3 we associate to each vertex v P V a spine bundle Θ v ĂĒ, which corresponds to a Seifert piece of the JSJ decomposition of the peripheral graph manifold, along with a pair of hyperbolic spaces K v and Ξ v that will figure into the HHS structure on Γ. The space K v is obtained via a quasimorphism constructed using the Seifert fibered structure following ideas in forthcoming work of the fourth author with Hagen, Russell, and Spriano [HRSS21], while Ξ v is coarsely obtained by coning off boundary components of the universal covers of the base 2-orbifold of this Seifert fibered manifold. We then appeal to the flat geometry of the fibers of E to construct and study certain projection mapsĒ and prove that various pairs of subspaces ofĒ have bounded projection onto each other (Proposition 3.19).
In §4, we begin assembling the combinatorial objects necessary to apply the HHG criterion from [BHMS20], which involves both combinatorial and geometric aspects. The first step involves the construction of a natural flag complex X containing the union of the Bass-Serre trees, together with appropriate "subjoins" with the union of all K v , over v P V. Next, we use the geometry ofĒ to construct a certain graph W whose vertices are maximal simplices of X and on which Γ acts metrically properly and coboundedly. The remainder of this section is devoted to verifying the necessary combinatorial conditions as well as translating the facts about K v and Ξ v and the projections described above into proofs of the necessary geometric conditions. We note that in the combinatorial HHG setup, the complex X comes with its own hierarchy projections between the induced hyperbolic spaces (Definitions 4.9-4.10), which may be different than the projections to K v and Ξ v .
In §5 we prove our QI-rigidity result Theorem 1.7. The starting point is the hierarchical hyperbolicity of Γ provided by Theorem 1.4, as it gives access to the results and arguments in [BHS21] about the preservation of quasi-isometrically embedded flats. Every collection of pairwise orthogonal hyperbolic spaces in an HHG determines a natural product subspace, with the maximal standard quasi-isometrically embedded flats (or orthants) arising inside such subspaces as products of quasi-lines in a maximal collection of pairwise orthogonal hyperbolic spaces of the HHG. Theorem A of [BHS21] states that a quasi-isometry of an HHS preserves the structure of its quasi-flats and takes any maximal quasi-flat within bounded Hausdorff distance of the union of standard maximal orthants. The maximal quasi-flats in the HHG structure on s E, namely the 2-dimensional flats indicated in Corollary 1.5, are encoded by certain strip bundles that, roughly, correspond to flats in the peripheral graph manifolds. We use the preservation of the maximal quasi-flats to derive coarse preservation of these strip bundles, which we then upgrade to coarse preservation of the fibers ( §5.1). By using tools of flat geometry from [BL18,DELS18], we then show any quasi-isometry induces an affine homeomorphism of any fiber to itself ( § §5.2-5.3) and moreover that this assignment is injective ( §5.4). Finally, we show this association is an isomorphism by proving ( §5.5) that every affine homeomorphism of a fiber induces an isometry and hence quasi-isometry ofĒ. Quasi-isometric rigidity and its algebraic consequence Corollary 1.8 are then easily obtained in §5.6.

Setup: The groups and spaces
Here we briefly recall the basic set up from [DDLS21] which we will use throughout the remainder of the paper. We refer the reader to Sections 2 and 3 of that paper for details and precise references.
2.1. Flat metrics and Veech groups. Fix a closed surface of genus at least 2, a complex structures X 0 (viewed as a point in the Teichmüller space T pSq), and a nonzero holomorphic quadratic differential q on pS, X 0 q. Integrating a square root of q determines preferred coordinates on pS, X 0 q for q which defines a translation structure (in the complement of the isolated zeros of q). We also write q for the associated flat metric defined by the half-translation structure (though the metric only determines the half-translation structure or quadratic differential up to a complex scalar multiple). This metric is a non-positively curved Euclidean cone metric, with cone singularities at the zeros of q. The orbit of pX 0 , qq under the natural SL 2 pRq action on quadratic differentials projects to a Teichmüller disk, D " D q Ă T pSq, which we equip with its Poincaré metric ρ. The circle at infinity of D is naturally identified with the projective space of directions, P 1 pqq, in the tangent space of any nonsingular point of q. For α P P 1 pqq, we write Fpαq for the singular foliation by geodesics in direction α.
We assume that the associated Veech group G " G q is a lattice-recall that G can be viewed as the stabilizer in the mapping class group of S of D as well as the affine group of q, and the lattice assumption is equivalent to requiring the quotient orbifold D{G to have finite ρ-area. The parabolic fixed points in the circle at infinity form a subset we denote P Ă P 1 pqq. This subset corresponds precisely to the completely periodic directions for the flat metric q; that is, the directions α for which the foliation Fpαq decomposes S into cylinders foliated by q-geodesic core circles. The boundaries of these cylinders are q-saddle connections (q-geodesic segments connecting pairs of cone points, with no cone points in their interior), and by the Veech Dichotomy, every saddle connection is in a direction in P. We let tB α u αPP denote any G-invariant, 1-separated set of horoballs in D and let D " D ď αPP intpB α q be the G-invariant subspace obtained by removing these horoballs. We writeρ for the induced path metric onD. Finally, we let p : D ÑD be the G-equivariant quotient obtained by collapsing each horoball B α to a point, for α P P. There is a natural path metricρ onD so that p is 1-Lipschitz and is a local isometry at every point not in one of the horoballs.
We will also make use of the closest point projection to the horoball for each α P P.

2.2.
The bundles E andĒ. For each point X P D, we let q X denote the associated flat metric or quadratic differential (defined up to scalar multiplication) on S. The space of interest E is a bundle over D, for which the fiber E X over X P D is naturally identified with the universal cover r S of S, equipped with the pull-back complex structure X and quadratic differential/flat metric q X . We write B α " π´1pB α q for α P P.
For any X, Y P D, the Teichmüller map between these complex structures has initial and terminal quadratic differentials q X and q Y (up to scalar multiple) and this map lifts to a canonical affine map between the fibers f Y,X : E X Ñ E Y . These maps satisfy f Z,X " f Z,Y f Y,X for all X, Y, Z P D, and for any X P D, assemble to a map f X : E Ñ E X defined by f X pyq " f X,πpyq pyq. Moreover, for any X, Y P D, f Y,X is e ρpX,Y q -bi-Lipschitz. We use the maps f X,X0 to identify P 1 pqq -P 1 pq X q for all X P D.
The fiber over X 0 is denoted E 0 " E X0 and the maps f 0 " f X0 : E Ñ E 0 and π : E Ñ D are projections on the factors in a product structure E -DˆE 0 -Dˆr S. For x P E, we write D x " f´1 πpxq pxq, which is just the slice Dˆtf 0 pxqu in the product structure. The affine maps f Y,X sends the cone points Σ X of E X to the cone points Σ Y of E Y . Consequently, the union of all singular points We give the space E a singular Riemannian metric d which is the flat metric on each fiber E X and the Poincaré metric on each diskD x so that at each smooth point of intersection, the tangent planes are orthogonal. The singular locus of this metric is precisely Σ. Each disk D x is isometrically embedded since π is a 1-Lipschitz map, and hence restricts to an isometry π| Dx : D x Ñ D. The metric on E Σ is in fact a locally homogeneous metric, modeled on a four-dimensional, Thurston-type geometry; see [DDLS21,§5].
The extension group Γ acts on E by bundle maps with the kernel π 1 S ă Γ of the projection to G acting trivially on D and by covering transformation on each fiber E X . We setĒ " π´1pDq Ă E, and writeπ :Ē ÑD. When convenient to do so, we put "bars" over objects associated toD orĒ, e.g.D x " D x XĒ,p :D ÑD, etc. In particular, we writed for the induced path metric onĒ Ă E, induced from the metric on E described above.
For any α P P, the closest point projection c α : D Ñ B α has a useful "lift" for any x PĒ. That is, f α maps each fiber E X via the map f Y,X to E Y , where Y " c α pXq is the image of the closest point projection to B α of X in D.
2.3. The hyperbolic spaceÊ. The quotient p : D ÑD is the descent of a quotient P : E ÑÊ which we now describe. First, for each α P P, the foliation Fpαq lifts to a foliation on E 0 in direction α, and hence on any fiber E X by push-forward via the map f X,X0 , also in direction α (via the identification P 1 pqq -P 1 pq X q).
There is a natural transverse measure coming from the flat metric on X. Given α P P, we fix some X α P BB α and let T α be the dual simplicial R-tree to this measured foliation in direction α on E Xα , and we let t α : E Ñ T α be the composition of the leafspace projection E Xα Ñ T α with the map f Xα : E Ñ E Xα . Now we define P : E ÑÊ to be the quotient space obtained by collapsing the subset B α to T α via t α | Bα for each α P P. We also writeP " P |Ē :Ē ÑÊ. The maps P andP descend to the maps p andp, and the map π determines mapsπ andπ, which all fit into the following commutative diagram. Theorem 2.1. There is a Gromov hyperbolic path metricd onÊ so thatP :Ē ÑÊ is 1-Lipschitz and is a local isometry at every point x PĒ´BĒ. Furthermore, for every α P P, ‚ The induced path metric on P pBB α q " T α is the R-tree metric determined by the transverse measure on the foliation of E Xα in direction α. ‚ The subspace topology on T α ĂÊ agrees with the R-tree topology on T α .
Remark 2.2. The underlying simplicial tree T α is precisely the Bass-Serre tree dual to the splitting of π 1 S defined by the cores of the cylinders of Fpαq on S.
For each x P E, we denote the image of D x inÊ byD x , which is obtained by collapsing B α X D x to a point, for each α P P. Consequently,π|D x :D x ÑD is a bijection, and so eachD x , with its path metric, is isometric toD and isometrically embedded inÊ. We call objects in E,Ē, andÊ vertical if they are contained in a fiber of π,π, orπ, respectively, and horizontal if they are contained in D x ,D x , or D x , for some x P E,Ē.
2.4. Vertices, spines, and spine bundles. We will write V ĂÊ for the union over all α P P of all vertices of T α . We will simultaneously view V as both a subset ofÊ and abstractly as an indexing set that will be used in sections §3-4 to develop an HHS structure onĒ. Since each vertex belongs to a unique tree, and since the trees are indexed by α P P, we obtain a map α : V Ñ P so that v is a vertex of T αpvq . For convenience, we also write B v " B αpvq , BB v " BB αpvq , etc for each v P V, and write c v " c αpvq for the ρ-closest point projection D Ñ B v .
For v, w P V, we write v w if αpvq " αpwq. Then define d tree pv, wq P Z ě0 Y t8u to be the combinatorial (integer valued) distance in the simplicial tree T αpvq " T αpwq when v w (as opposed to the distance from the R-tree metric) and to equal 8 when v ∦ w.
Given α P P, X P D, and v P T α , the v-spine in E X is the subspace X is the union of the saddle connections on the fiber E X in direction α that project to v by t α . When d tree pv, wq " 1 (and hence v, w are adjacent in the same tree T α ) there is a unique component of E X pθ v X Y θ w X q whose closure is an infinite strip, Rˆra, bs, that covers a maximal cylinder in the quotient E X {π 1 S " pS, X, q X q in the direction α. We let Θ v X be the union of θ v X and all such strips defined by w P T α with d tree pv, wq " 1. We call These spaces are bundles over BB v which we call, respectively, the v-spine bundle and the thickened v-spine bundle.
2.5. Schematic of the spaceĒ and its important pieces. Figure 1 is a cartoon of the bundleĒ over the truncated Teichmüller diskD. We have tried to highlight some of the key features ofĒ which are relevant to this paper.
(a) The stabilizer of a horoball based at a point β P P is virtually cyclic, generated by a multitwist τ α acting as a parabolic on D. The base point X β on the horocycle based at β and its image are shown. (b) The bundle over the boundary horocycle based at β is shown. This is the universal cover, BB β , of a graph manifold which is the mapping torus of τ β . Two fibers E X β and E τ β pX β q are shown with the effect on a part of a spine (in green) in some other direction illustrating the sheering in strips after applying τ β . (c) This is another horoball in some direction α, with the chosen basepoint X α and its horocycle BB α . (d) The spine θ v Xα in direction α is shown in red, corresponding to a vertex v P T α . The thickened spine Θ v Xα is indicated in lavender. Spines for vertices of T α adjacent to v meet Θ v Xα along lines in BΘ v Xα and are shown in various other colors. (e) The restriction of t α : E Ñ T α to E Xα collapses each spine θ w Xα or strip in direction α to the corresponding vertex w or edge the Bass-Serre tree T α . The spaceÊ is formed by collapsing B α to T α via t α . 2.6. Some technical lemmas and coarse geometry. Here we briefly recall some basic facts about the setup above proved in [DDLS21] as well as some useful coarse geometric facts. The first fact is the following; see [DDLS21, Lemma 3.4].
Lemma 2.3. There exists a constant M ą 0 such that for each v P V and X P BB v , every saddle connection in θ v X has length at most M and every strip in Θ v X has width at most M . In particular, for points X P BB α , the saddle connections and strips of E X in direction α P P have, respectively, uniformly bounded lengths and widths.
Every connected graph can be made into a geodesic metric space by locally isometrically identifying each edge with a unit interval. We will need the following well-known result (for a proof of this version, see [DDLS21, Proposition 2.1]).
Proposition 2.4. Let Ω be a path metric space and Υ Ă Ω an R-dense subset for some R ą 0. For any R 1 ą 3R, consider a graph G with vertex set Υ such that: ‚ all pairs of elements of Υ within distance 3R are joined by an edge in G, ‚ if an edge in G joins points w, w 1 P Υ, then d Ω pw, w 1 q ď R 1 . Then the inclusion of Υ into Ω extends to a quasi-isometry G Ñ Ω.
The following criterion for a graph to be a quasi-tree is well-known, and an easy consequence of Manning's bottleneck criterion [Man05]. We include a proof for completeness.
Proposition 2.5. Let X be a graph, and suppose that there exists a constant B with the following property: For each pair of vertices w, w 1 there exists an edge path γpw, w 1 q from w to w 1 so that for any vertex v on γpw, w 1 q, any path from w to w 1 intersects the ball of radius B around v. Then X is quasi-isometric to a tree, with quasi-isometry constants depending on B only.
Proof. We check that [Man05, Theorem 4.6] applies; that is, we check the following property. For any two vertices w, w 1 P X, there is a midpoint mpw, w 1 q between w and w 1 so that any path from w to w 1 passes within distance B 1 " B 1 pBq of mpw, w 1 q. (The uniformity in the quasi-isometry comes from the proof of Manning's theorem, see [Man05, page 1170].) Consider any geodesic α from w to w 1 , and let m " mpw, w 1 q be its midpoint. We will show that m lies within distance 2B`1 of a vertex of γ " γpw, w 1 q, so that we can take B 1 " 3B`1.
Indeed, suppose by contradiction that this is not the case. Let w " w 0 , . . . , w n " w 1 be the vertices of γ (in the order in which they appear along γ), and let d i " dpw, w i q, so that |d i`1´di | ď 1. Each w i lies within distance B of some point p i on α which must satisfy dpp i , mq ě B`1. In particular, we have that every d i satisfies either d i ď dpw, w 1 q{2´1 or d i ě dpw, w 1 q{2`1. Since d 0 " 0 and d n " dpw, w 1 q, we cannot have |d i`1´di | ď 1 for all 0 ď i ď n´1, a contradiction.
We end with a few definitions from coarse geometry which may not be completely standard but will appear in the next two sections. Given two metrics d and d 1 on a set X, we say that d is coarsely bounded by d 1 if there exists a monotone function N : r0, 8q Ñ r0, 8q so that dpx, yq ď N pd 1 px, yqq, for all x, y P X. If d is coarsely bounded by d 1 and d 1 is coarsely bounded by d, we say that d and d 1 are coarsely equivalent. An isometric action of a group H on a metric space Y is metrically proper if for any R ą 0 and any point y P Y , there are at most finitely many elements h P H for which h¨Bpy, Rq X Bpy, Rq ‰ H. For proper geodesic spaces, this is equivalent to acting properly discontinuously. If there exists y, R so that H¨Bpy, Rq " Y , then we say that the action is cobounded, and for proper geodesic metric spaces this is equivalent to acting cocompactly.

Projections and vertex spaces
An HHS structure on a metric space consists of certain additional data, most importantly a collection of hyperbolic spaces together with projection maps to each space. For the HHS structure that we will build on (Cayley graphs of) Γ, the hyperbolic spaces will (up to quasi-isometry) be the spaceÊ from [DDLS21] (see §2.3) and the spaces K v and Ξ v introduced in this section, where v varies over all vertices of the trees T α . Morally, the projections will be given by the maps Λ v and ξ v that we study below. However, to prove hierarchical hyperbolicity we will use a criterion from [BHMS20] which does not require actually defining projections, but nevertheless provides them. Still, the maps Λ v and ξ v will play a crucial role in proving this criterion applies.
We will establish properties of Λ v and ξ v that are reminiscent of subsurface projections or of closest-point projections to peripheral sets in relatively hyperbolic spaces/groups; these are summarized in Proposition 3.19. Essentially, these same properties would be needed if we wanted to construct an HHS structure on Γ directly without using [BHMS20].
From a technical point of view, we would like to draw attention to Lemma 3.13, which is the crucial lemma that ensures that the projections behave as desired and that various subspaces have bounded projections. Roughly, the lemma says that closest-point projections to a spine do not vary much under affine deformations.
In what follows, we will write d Θ v and d BBα for the path metrics on Θ v and BB α induced fromd. Using the map f α :Ē Ñ BB α , it is straightforward to see that d BBα is uniformly coarsely equivalent to the subspace metric fromd: in fact, d ď d BBα ď edd. The same is true for d Θ v , which follows from the fact that the inclusion of Θ v into BB α is a quasi-isometric embedding with respect to the path metrics (see below).
Associated to each v P V we will be considering two types of projections. These projections have a single projection Π v :Σ Ñ Θ v as a common ingredient. It is convenient to analyze Π v via an auxiliary map which serves as a kind of fiberwise closest point projection that survives affine deformations, and which we call the window map. We describe the two types of projections restricted to Θ v , as well as the target spaces of said projections, in §3.1 and §3.3, where we also explain some of their basic features. Next we define the window map and prove what is needed from it. Finally, we define Π v and prove the key properties of the associated projections.
3.1. Quasimorphism distances. For each v P V, we will use ideas from work-inprogress of the fourth author with Hagen, Russell, and Spriano [HRSS21] to define a map where K v is a discrete set quasi-isometric to R. The key properties of this map are given by the next proposition. We note that the proposition and Lemma 3.6 can be used as black-boxes (in particular, the definitions of λ v and K v are never used after we prove those results).
Proposition 3.1. There exists K 1 ą 0 such that, for each v P T p0q α Ă V, there exist a space K v that is pK 1 , K 1 q-quasi-isometric to R and a map λ v : Θ v Ñ K v satisfying the following properties: (1) λ v is K 1 -coarsely Lipschitz with respect to the path metric on Θ v .
(2) For any x P BΘ v , if x,α " D x X BB α then λ v p x,α q is a set of diameter bounded by K 1 .
(3) For any v, w P V with d tree pv, wq " 1, λ vˆλw : Θ v X Θ w Ñ K vˆKw is a K 1 -coarsely surjective pK 1 , K 1 q-quasi-isometry with respect to the induced path metric on the domain. (4) (Equivariance) For any g P Γ and v P T p0q α there is an isometry g : K v Ñ K gv and for all x P Θ v we have λ gv pgxq " gλ v pxq.
The sets x,α in item (2) are certain lines whose significance is explained below.
Remark 3.2. An earlier version of this paper used work of Kapovich and Leeb to construct the spaces K v and maps λ v , resulting in a weaker version of this proposition which did not include the last, equivariance condition. Consequently Γ could only be shown to be an HHS, rather than an HHG. The ideas from [HRSS21] were crucial in this extension.
To explain the proof of the proposition, it is useful to review some background on graph manifolds, which we do now.
Graph manifolds and trees. Recall that a graph manifold is a 3-manifold that contains a canonical finite union of tori (up to isotopy), so that cutting along the tori produces a disjoint union of Seifert fibered 3-manifolds, called the Seifert pieces. Seifert fibered 3-manifolds are compact 3-manifolds foliated by circle leaves; see [JS79].
The universal cover of a graph manifold decomposes into a union of universal covers of the Seifert pieces glued together along 2-planes (covering the tori). The decomposition is dual to a tree, and the universal covers of the Seifert pieces are the vertex spaces. For any Seifert fibered space, its universal cover is foliated by lines, the lift of the foliation by circles, and we refer to the leaves simply as lines in the universal cover.
Horocycles and bundles. Next we describe the specific graph manifolds that are relevant for our purposes.
Let G α ă G denote the stabilizer of B α , for each α P P. This has a finite index cyclic subgroup G 0 α generated by a multitwist, xτ α y " G 0 α ă G α ; see e.g. [DDLS21, §2.9]. The preimage of G α in Γ is the π 1 S-extension group Γ α of G α , and we likewise denote by Γ 0 α ă Γ α the extension group of G 0 α . The action of Γ α on BB α is cocompact, and BB α {Γ α has a finite sheeted (orbifold) covering by BB α {Γ 0 α , which is the graph manifold mentioned in the introduction.
Consider the surface S with the flat metric q Xα , so that pS, X α , q Xα q " E Xα {π 1 S. The multitwist τ α is an affine map that preserves the cylinders in direction α, acting as a power of a Dehn twist in each cylinder and as the identity on their boundaries. The union of the boundaries of the cylinders are spines (deformation retracts) for the subsurfaces that are the complements of the twisting curves (core curves of the cylinders). Consequently, τ α is the identity on these spines. The homeomorphism τ α induces a homeomorphism on the subsurface obtained by cutting open S along a core curve of each cylinder. Each such induced homeomorphism is the identity on the corresponding spine, and is thus isotopic to the identity relative to the spine. The mapping torus of each subsurface is a product of the subsurface times a circle, and embeds in the the mapping torus BB α {Γ 0 α of τ α . These sub-mapping tori are the Seifert pieces for the graph manifold structure on BB α {Γ 0 α . The lifted graph manifold decomposition of BB α corresponds to T α . That is, for each v P T p0q α , there is a vertex space contained in Θ v and containing θ v . In fact, with respect to the covering group, Θ v is an invariant, bounded neighborhood of the vertex space and θ v is an equivariant deformation retraction of that space. We let Γ v ă Γ α denote the stabilizer of Θ v in Γ α and Γ v0 ă Γ 0 α the stabilizer in Γ 0 α . The suspension flow on the mapping torus BB α {Γ 0 α restricted to each spine defines circle leaves of the corresponding Seifert piece; that is, flow lines through any point on the spine are precisely the circle leaves. In the universal covering BB α , the lifted flowline through a point x P BB α is a lifted horocycle, x,α " D x X BB α . Thus, for any vertex v and any x P θ v , x,α is a line for the vertex space corresponding to v. We note that not only does Γ v0 preserve this set of lines, but so does Γ v .
For any x P θ v , the stabilizer in Γ v0 of x,α is generated by a lift g v of τ α . Therefore, the quotient Θ v {Γ v0 is homeomorphic to a product, If we do not care about the particular point X over which we take the fiber, we simply write S v for the Since E X is a copy of the universal cover of S, we can consider S v as a subsurface of S (embedded on the interior) and π 1 S v is its fundamental group inside π 1 S (up to conjugacy).
is an orbifold cover sending circles to circles making Θ v {Γ v into a Seifert fibered orbifold (some of the Seifert fibers may be part of the orbifold locus) that also (orbifold)-fibers over the circle (with finite order monodromy).
for the Seifert fibration to the quotient 2-orbifold. Further write 1 pO v q for the induced homomorphism of the Seifert fibration and for the induced homomorphism from the fibration over the circle. Because g v acts as translation on the line x,α for x P θ v , it represents a loop that traverses a circle in the Seifert fibration, which is thus also a suspension flowline for the fibration over the circle. Thus we have ν v pg v q " 0 and φ v pg v q ‰ 0. To complete the picture, we note that restricting Finally, note that for any w adjacent to Remark 3.3. One caveat about the lines for the vertex spaces: flowlines through points not on a spine are not lines of any vertex space. In fact, they are not even uniformly close to lines for any vertex space.
Constructing the map. Here we define K v and λ v and prove the main properties we will need about them. We require a little more set up first. We choose representatives of the Γ-orbits of vertices, is a finite generating set for Γ v . The Γ v -translates of ∆ v define a tiling of Θ v , and the map sending every point of g∆ v to g P Γ v is a quasi-isometry by the Milnor-Schwarz Lemma. We denote this map as r We note that any word metric on Γ v defines a "word metric" on each coset gΓ v , for g P Γ (elements are distance 1 if they differ by right multiplication by an element of the generating set). We can push the tiling forward by g to a Γ gv " gΓ v g´1invariant tiling of Θ gv (if g P Γ v , this is precisely the given tiling of Θ v ). For any element g 1 P gΓ v , the map that sends every point in g 1 ∆ v to g 1 defines a quasiisometry r λ gv : Θ gv Ñ gΓ v which is Γ gv -equivariant, with the same quasi-isometry constants. If g 1 P Γ and x 1 P ∆ v , then for all g P Γ On the other hand, any w P Γ¨v and x P Θ w have the form w " g 1 v and x " g 1 x 1 for some g 1 P Γ and x 1 P ∆ v . Thus, for any g P Γ, the equation above becomes (2) r λ gw pgxq " r gλ w pxq Having carried out the construction above for each v P V 0 and each vertex in its orbit, we have maps r λ w from Θ w to a coset of a vertex stabilizer from V 0 for every w P V, so that equation (2) holds for every x P Θ w , and g P Γ.
Next, recall that a homogeneous quasimorphism (with deficiency D) from a group H to R is a map ψ : H Ñ R such that for all h, h 1 , h 2 P H and n P Z we have ψph n q " nψphq and |ψph 1 h 2 q´ψph 1 q´ψph 2 q| ď D.
Proof. Let w 1 , . . . , w r be Γ v -orbit representatives of the vertices adjacent to v. Here r is the number of boundary components of O v , so that ν v pg w1 q, . . . , ν v pg wr q are peripheral loops around the r distinct boundary components of O v . Since π orb 1 O v is the fundamental group of a hyperbolic 2-orbifold with non-empty boundary, appealing to [HO13, Theorem 4.2], which applies to π orb 1 O v and its subgroups xν v pg wi qy in view of [DGO17, Corollary 6.6, Theorem 6.8], one can find a homogeneous quasimorphism should also be applicable to construct such quasimorphisms). Set s 0 " 1{φ v pg v q, and for each i " 1, . . . , r, set s i " s 0 φ v pg wi q, and then define As a linear combination of homogeneous quasimorphisms, ψ v is a homogeneous quasimorphism. Since On the other hand, for any j " 1, . . . , r we have proving the lemma.
According to [ABO19, Lemma 4.15], there is an (infinite) generating set for Γ v so that with respect to the resulting word metric, the quasimorphism ψ v : . For any g P Γ, define K gv to be the coset gΓ v with this generating set so that r λ gv defines a map Carrying this out for every v P V 0 , (2) implies (3) λ gw pgxq " gλ w pxq for all w P V and x P Θ w , and g P Γ.
Before we proceed to the proof of Proposition 3.1, observe that Γ gv " gΓ v g´1 acts isometrically on gΓ v with respect to any generating set, and thus we can use this to define a generating set for the conjugate so that (any) orbit map is an isometry; in fact, this will just be a conjugate of the generating set for Γ v . In particular, when convenient we will identify K gv isometrically with the conjugate gΓ v g´1 via such an orbit map. Conjugating the quasimorphisms ψ v from the lemma, for v P V 0 , we obtain uniform quasi-isometries for all w P V, which for an appropriate choice of identification of K w with a conjugate of some Γ v , v P V 0 , is a quasimorphism (with uniformly bounded deficiency).
Proof of Proposition 3.1. From the discussion above and Equation (3), we immediately see that item (4) of the proposition holds. Next, observe that by adding finitely many generators to the infinite generating set of Γ v0 for any v 0 P V 0 , changes K v0 by quasi-isometry. On the other hand, the finite generating set described in Equation (1) for v 0 P V 0 makes r λ v0 a quasiisometry. Thus, adding these generators to the infinite generating set does not change the quasi-isometry type of K v0 , but clearly makes λ v0 coarsely Lipschitz. Therefore, λ v is uniformly coarsely Lipschitz for all v P V, and hence item (1) holds for all v P V.
To prove item (2), let v P V and x P BΘ v . Then x P θ w , for some w P V adjacent to v. As discussed above, we view K v and K w as conjugates Γ v and Γ w of groups Γ v0 and Γ w0 , respectively, for v 0 , w 0 P V 0 , equipped with their conjugated infinite generating sets. Let ψ v : K v Ñ R and ψ w : K w Ñ R be the associated uniform quasi-isometric homogeneous quasimorphisms. The element g w P Γ v stabilizes x,α acting by translation on it, and by construction, ψ v pg w q " ψ v pg n w q " 0 for all n P Z. It follows that every orbit of xg w y acting on K v is uniformly bounded. Indeed, if D is the deficiency of ψ v , then for any g P K v , we have |ψ v pg n w gq´ψ v pgq| " |ψ v pg n w gq´ψ v pgq´ψ v pg n w q| ď D and therefore g n w g and g are uniformly bounded distance apart in K v (since ψ v is a uniform quasiisometry). Now, since g n w v " v, by item (4) of the proposition we have λ v pg n w xq " λ g n w v pg n w xq " g n w λ v pxq, and since g n w λ v pxq is uniformly close to λ v pxq, it follows that λ v sends the xg w yorbit of x to a uniformly bounded set. Since this orbit is R-dense in x,α for some uniform R ą 0, and since λ v is uniformly coarsely Lipschitz (by item (1)) we see that λ v p x,α q has uniformly bounded diameter. This proves item (2).
For item (3), we continue with the assumptions on v, w as above. Note that since ψ v pg n v q " n, using again the fact that ψ v is a uniform quasi-isometric homogeneous quasimorphism to R, it follows that for any x P Θ v , the map n Þ Ñ λ v pg n v xq is a uniformly coarsely surjective, uniform quasiisometry Z Ñ K v . Since every orbit of xg w y on K v is uniformly bounded, it follows that for all n, m P Z, the two points λ v pg n v g m w xq " g m w λ v pg n v xq and λ v pg n v xq are uniformly close to each other. Likewise, λ w pg n v g m w xq and λ w pg m w xq are also uniformly close to each other. But this means that λ vˆλw pg n v g m w xq and pλ v pg n v xq, λ w pg m w xqq are uniformly close, and thus pn, mq Þ Ñ λ vˆλw pg n v g m w xq is a uniformly coarsely surjective, uniform quasiisometry Z 2 Ñ K vˆKw .
On the other hand, the assignment pn, mq Þ Ñ g n v g m w x defines a uniform quasiisometry Z 2 Ñ Θ v X Θ w since xg w yˆxg v y -Z 2 acts cocompactly on Θ v X Θ w (with uniformity coming from the fact that there are only finitely many Γ-orbits of pairs pv, wq of adjacent vertices). Combining these two facts, together with the fact that λ v and λ w are uniformly coarsely Lipschitz, it follows that is a uniformly coarsely surjective, uniform quasiisometry. This proves item (3), and completes the proof of the proposition.
3.2. A technical lemma. The goal of this subsection is to prove Lemma 3.6, whose relevance will only be clear in §4. We prove it here since we have now established the setup for its proof.
We recall that for each X for some X P BB α (and α " αpvq) and the slices txuˆR (more precisely, the level sets of pµ v q´1pxq Ă Θ v ) are lines for Θ v . These lines project to circle fibers in Θ v {Γ v and we may assume they contain all the lines Lemma 3.5. The map µ vˆλv : Θ v Ñ r S vˆKv is a uniform, Γ v -equivariant quasi-isometry with uniformly dense image. Moreover, the constant K 1 from Proposition 3.1 can be chosen so that for any v P V and s P K v , the subspace has the property that µ vˆλv pM psqq has uniformly bounded Hausdorff distance to the slice r S vˆt su, and furthermore M psq nontrivially intersects every line of Θ v .
We note that the intersection of M psq with each line of Θ v is necessarily a uniformly bounded diameter set by the uniform bounded Hausdorff distance condition.
Proof. All constants will be independent of the specific vertex v, so we drop it from the notation. We write d for all path-metric distances in what follows (the location of points will determine which metric is being used). Products are given the L 1 metric for convenience. We further let K be the maximum of the coarse Lipschitz constants of µ, ρ, λ and the biLipschitz constant of µˆρ, and assume, as we may, that K ě 2. From the proof of Proposition 3.1(3), if x is any point of a line of Θ, then n Þ Ñ λ v pg n v xq is a uniformly coarsely surjective, uniform quasi-isometry from Z to K. Therefore, λ " λ v is a uniformly coarsely surjective, uniform quasi-isometry from any line of Θ to K. We further assume that the coarse surjectivity constants and quasi-isometry constants are also all taken to be K.
We are left to consider the case that the maximum in (5) is realized by dpρpxq, ρpyqq, which thus satisfies dpρpxq, ρpyqq ě K 4 dpµpxq, µpyqq. Let z P Θ be such that ρpzq " ρpxq and µpzq " µpyq. Since µpzq " µpyq, z and y lie on a line, and since the restriction of λ to this line is a pK, Kq-quasi-isometry, we have K 2 dpρpxq, ρpyqq`K Combining the previous two sets of inequalities and the triangle inequality, we have dpλpxq, λpyqq ě dpλpzq, λpyqq´dpλpzq, λpxqq K 2 dpρpxq, ρpyqq´2K. Combining this inequality with (5) where we have assumed the maximum is realized by dpρpxq, ρpyqq, we obtain dpµˆρpxq, µˆρpyqq ď 2dpρpxq, ρpyqq . which provides the required upper bound. This completes the proof of the first claim of the lemma.
For the second claim of the lemma, we now increase K 1 from Proposition 3.1 if necessary, so that K 1 ě K. Observe that That is, pµˆλq˝pµˆρq´1 sends the line txuˆR to txuˆK, for any x P r S. As already noted at the start of the proof, restricting to this line, λ is K-coarsely Lipschitz and K-coarsely onto. Therefore, for any s P K and x P r S, there exists t so that λppµˆρq´1px, tqq is within K 1 ě K of s. Thus, for any line of Θ, the λ-image nontrivially intersects N K1 psq, and hence this line nontrivially intersects M psq. By definition, µˆλ maps M psq into r S vˆN K1 psq, and by the previous sentence, every point of r Sˆtsu is within K 1 of some point of µˆλpM psqq. Thus, λˆµpM psqq has Hausdorff distance at most K 1 from r Sˆtsu, as required.
As mentioned above, the following technical lemma will be needed in §4. In the statement M psq is as defined by Lemma 3.5.
Lemma 3.6. There is a function N " N K : r0, 8q Ñ r0, 8q with the following property. Suppose that v, v 1 , v 2 P V are so that d tree pv, v i q " 1. Then for each s P K v and t i P K vi we havē dpM psq X M pt 1 q, M psq X M pt 2 qq ď N pdpM pt 1 q, M pt 2 qqq.
Proof. It suffices to prove the lemma withd replaced by the path metric d BBα on BB α , since they are uniformly coarsely equivalent. In fact, it will be convenient to consider the path metric d 0 on the union of the three vertex subspaces which is also uniformly coarsely equivalent since each vertex space uniformly quasiisometrically embeds in BB α . In this subspace, we will actually prove that the two distances are uniformly comparable. Now, for each i " 1, 2 the uniform quasi-isometry µ viˆλvi : S vi be the closest point projection, and then set (6)). This map is a uniformly coarsely Lipschitz, coarse retraction of Θ vi onto Θ vi X Θ v . Moreover, this sends M pt i q, which is uniformly close to the pµ viˆλvi q´1-image of r S viˆt t i u, to a uniformly bounded neighborhood of M pt i q X Θ v . Consequently, Claim 3.7. The quasi-isometry µ vˆλv maps M pt i q X Θ v within a uniformly bounded Hausdorff distance of the slice tz i uˆK v Ă r S vˆKv , for each i " 1, 2, where pz i , t 1 i q is a point in the µ vˆλv -image of M psq X M pt i q. Assuming the claim, we note then that again with uniform constants. Combining this coarse equation with (7) we get the required uniform estimate Fix i " 1 or 2 and we prove the claim. Since λ vˆλvi is K 1 -coarsely surjective (Proposition 3.1(3), there exists some point y i P Θ v X Θ vi with pλ v py i q, λ vi py i qq within distance K 1 of ps, t i q P K vˆKvi . Therefore, y i P M psq X M pt i q and we set µ vˆλv py i q " pz i , t 1 i q.
Next, we observe that λ vi is uniformly coarsely constant on any line of Θ v contained in Θ v X Θ vi by Proposition 3.1(2) and uniformly coarsely Lipschitz by Proposition 3.1(1). Hence, the line a neighborhood of uniformly bounded radius of K vˆt t i u. Therefore, any point in the image of the line in K vˆKvi lies uniformly close to a point in λ vˆλvi pM pt i q X Θ v q by Proposition 3.1(3) (which guarantees that any point in K vˆt t i u is K 1 -close to a point in the image of the subspace Θ v X Θ vi ). Therefore any point in the line lies uniformly close to some point in M pt i q X Θ v since λ vˆλvi is a uniform quasi-isometry again by Proposition 3.1(3).
On the other hand, Lemma 3.5 implies µ viˆλvi pM pt i qq is uniformly bounded Hausdorff distance from the slice r S viˆt t i u Ă r S viˆKvi . Moreover, since M pt i q meets every line of Θ vi (Lemma 3.5 again), it follows that In particular, M pt i q X Θ v is itself a uniform quasi-line and consequently lies within a uniformly bounded neighborhood of the line pµ v q´1pz i q. Since this line maps within within a uniformly bounded Hausdorff distance of the slice tz i uˆK v in r S vˆKv by µ vˆλv , we see that M pt i q X Θ v does as well 3.3. Coned-off surfaces. For v P V, we define Ξ v to be the graph whose vertices are all w P V so that d tree pv, wq " 1, and with edges connecting the pairs w, w 1 whenever Θ w X Θ w 1 ‰ H. As such, vertices w P V are in bijective correspondence with the boundary components of Θ v and there is an "inclusion" map that sends any point x P BΘ v to the vertex w for which x P Θ w . In light of the following lemma, we note that we could alternately define the edges of Ξ v in terms of subspaces lying within bounded distance of each other, and produce a space quasi-isometric to Ξ v .
where M is the bound on the width of a strip and length of a saddle connection from Lemma 2.3. Since the path metric d θ v on θ v is coarsely equivalent to the subspace metric, d θ v pp 1 , p 2 q is bounded in terms of r. The path metric on θ v is biLipschitz equivalent to the 1 -metric on the product θ v XvˆR . Since each edge of θ v Xv has definite length, there is a path from p 1 to p 2 in θ v obtained by concatenating boundedly many (in terms of r) paths α i with f Xv pα i q a saddle connection in θ v Xv . Since each α i is contained in some Θ w , we are done.
Proof. Let w " i v pxq and w 1 " i v pyq. Since d Θ v andd are path metrics, we havē dpx, yq ď d Θ v px, yq. By Lemma 3.8, x and y may be joined by a concatenation α 1¨¨¨αk of k ď N pd Θ v px, yqq`2 paths α j each of which lies in some Θ v XΘ wj , and where w " w 1 and w 1 " w k . For successive paths α j , α j`1 , the vertices w j , w j`1 are Lemma 3.10. Each Ξ v is uniformly quasi-isometric to a tree. In particular, there exists δ ą 0 so that each Ξ v is δ-hyperbolic. Moreover, Ξ v has at least two points at infinity.
Proof. We appeal to Proposition 2.5 and show that for any vertices w, w 1 of Ξ v there exists a path γpw, w 1 q so that any path from w to w 1 passes within distance 3 of each vertex of γpw, w 1 q.
First, note that Ξ v is isomorphic to the intersection graph of the collection of strips in Θ v 0 . For each strip we have a vertex, and for each saddle connection of the spine θ v 0 , there is an edge of Ξ v that connects the vertices corresponding to the strips that contain the saddle connection. For each cone point in the spine θ v 0 , there is also a complete graph on the vertices corresponding to strips that contain this cone point. This accounts for all edges (because intersections of strips either arise along saddle connections or single cone points), and we note that the closure of each edge of the first type separates Ξ v into two components.
Suppose w, w 1 P Ξ v are two vertices, and let x, x 1 P θ v 0 be points in the (boundaries of the) strips corresponding to w and w 1 , respectively, that are closest in Θ v 0 , and consider the geodesic in θ v 0 connecting these points, which is a concatenation of saddle connections σ 1 σ 2¨¨¨σn . For each 1 ď i ď n, let wȋ be the vertices corresponding to the two strips Aȋ that intersect in the saddle connection σ i . We can form an edge path γpw, w 1 q in Ξ v , containing w, w 1 , and the wì as vertices, since Aì X Aì`1 ‰ H. Observe that any path from x to x 1 must pass through the union Aì Y Aí , for each i, since x and x 1 lie in the closures of distinct components of Θ v 0 pAì Y Aí q. Now let w " w 0 , w 1 , . . . , w k " w 1 be the vertices of an edge path connecting w to w 1 in Ξ v . For any points in the strips corresponding to w and w 1 , respectively, it is easy to construct a path in Θ v 0 between these points that decomposes as a concatenation ν 1 ν 2¨¨¨νk so that ν j is contained entirely in the strip corresponding to w j . From the previous paragraph, this path must pass through Aì Y Aí , for each i " 1, . . . , n. It follows that for each 1 ď i ď n, the edge path must meet the union of the stars starpwì q Y starpwí q. Since these stars intersect, their union has diameter at most 3, and we are done.
We now show that Ξ v contains a quasi-geodesic line. Consider strips A i of Θ v 0 , for i P Z, such that for all i we have ‚ A i and A i`1 share a saddle connection; ‚ A i´1 and A i`2 lie on distinct components of the complement of the interior of A i Y A i`1 in Θ v 0 . The A i give a bi-infinite path in Ξ v , and we now show that this path is a quasigeodesic. Fix integers m, n and consider a geodesic γ in Ξ v from A m to A n (where we think of the strips themselves as vertices of Ξ v for convenience). Then for each m ă k ă n´1 we have that γ needs to contain a vertex vpkq which, regarded as a strip, intersects A k or A k`1 . Indeed, the interior of A k Y A k`1 separates A m from A n , and the sequence of vertices of γ corresponds to a connected union of strips containing A m and A n . Moreover, there is no strip intersecting both A k and A k 1 if |k´k 1 | ě 3, and in particular we have vpkq ‰ vpk 1 q if |k´k 1 | ě 4. These observations imply that γ contains at least tpn´m´2q{4u vertices, so that geodesics connecting A m to A n have length comparable to n´m, and the A i form a quasi-geodesic line as required.
3.4. Windows and bridges. For v P V, consider the setΣ v of points inΣ that are inside some v-spine, as well as those pointsΣ Rv that are outside every v-spine: For each Y PD we now define a window map Π v Y :Σ Ñ PpBΘ v Y q from cone points to the set of subsets of the boundary The second case, that ofΣ v , is handled slightly differently: where the distance is computed in the path metric on E Y (or equivalently on Θ v Y ). Finally, we extend window maps to arbitrary subsets by declaring Proof. This is immediate since Π v Y pxq is defined just in terms of f Y pxq " f Y pyq. The following gives a counterpoint to Lemma 3.11 for points inΣ v .
Lemma 3.12. There exists K 2 ą 0 such that for any v P V the following holds: If x, y PΣ v satisfy f pxq " f pyq and either (1) x and y are connected by a horizontal geodesic of length ď 1, or (2) x and y are contained in BB w for some w P V with αpwq ‰ αpvq, then d v X px, yq ď K 2 , where X " c v pπpxqq. Proof. Set X " c v pπpxqq and Y " c v pπpyqq. Since c v : D Ñ BB v is 1-Lipschitz and diampc v pBB w qq is uniformly bounded for all such w, either condition (1) or (2) gives a uniform bound K ą 0 on the distance between X and Y . Hence f X,Y is e K -biLipschitz. The distance between f Y pyq P θ v Y and its closest cone points Π v Y pyq in BΘ v Y is also uniformly bounded by 2M , by Lemma 2.3. The same holds for the distance between f X pxq and Π v X pxq. It follows that The The next lemma explains that the image of Π v Y is not so far from being a point. Lemma 3.13 (Window lemma). For any v P V, Y PD, and x PΣ Rv , the window Take any flat geodesic rf Y pxq, zs in E Y from f Y pxq to a cone point z P . The geodesic rf Y pxq, zs first meets in some cone point p. Since the total cone angle at p is at least 3π and the angle at p along the side of containing Θ v is exactly π, the last saddle connection δ in the geodesic rf Y pxq, ps Ă rf Y pxq, zs must make an angle of at least π with on one side. Hence any geodesic from f Y pxq to a cone point on that side of p must pass through p.
If both angles between δ and at p are at least π, then any geodesic from f Y pxq to passes through p. Hence p is the unique point in BΘ v Y closest to f Y pxq and Π v Y pxq " tpu is a cone point as required. Otherwise, consider the flat geodesic from f Y pxq to the adjacent cone point p 1 on the other side of p along . The last saddle connection of this geodesic must also make an angle with of at least π on one side. This cannot be the side containing p, or else the geodesic from f Y pxq to p would pass through p 1 contradicting our choice of p. Hence any geodesic from f Y pxq to a cone point on the opposite side of p 1 must pass through p 1 . Therefore Π v Y pxq is the saddle connection between p and p 1 , and we are done. See Figure 2.
The following lemma gives us partial control over the window for points in adjacent vertex spaces in the same Bass-Serre tree.
Lemma 3.14 (Bridge lemma). For any v, w P V with d tree pv, wq " 1, any Y P D, and any component U Ă E Y θ w Y not containing θ v Y , there exists a (possibly degenerate) saddle connection δ U Ă BΘ v Y with the following property: We call δ U the bridge for U in E Y . It is clear from the construction in the proof below that f Z,Y pδ U q is the bridge for f Z,Y pU q, for any Z PD.
Proof. Let U be as in the statement and W be the component of which are both bi-infinite flat geodesics in θ w Y . If γ U Xγ W " H, then there is a unique geodesic between them in θ w Y , and we take δ U to be the endpoint of this geodesic which lies along γ W . On the other hand, if γ U X γ W ‰ H, then their intersection is contained in the boundary of a strip along θ v Y and another along θ w Y . Two distinct strips in the same direction that intersect do so in either a single point or a single saddle connection, and hence γ U X γ W is a point or single saddle connection, and we call δ U . See Figure 3. Figure 3. Bridges, and the proof of Lemma 3.14. Now consider any point x PΣ with f Y pxq P U . Observe that x PΣ Rv so that Π v Y pxq falls under the first case of the window definition. Further, any flat geodesic from f Y pxq to Θ v Y must pass through both γ U and γ W , and hence must pass through δ U Ă BΘ v Y . It follows that Π v Y pxq Ă δ U , as required. The following is an easy consequence of the previous lemma.
αpvq between v and w with d tree pu, vq " 1, and a component U Ă E Y zθ u Y whose closure contains Θ w Y . Setting δ v Y pwq " δ U and applying Lemma 3.14 completes the proof.
The next corollary is similar.
is contained in the bridge δ U for U by Lemma 3.14. Otherwise θ u Y X θ w Y ‰ H, and we claim there is a cone Indeed, if θ u Y X θ w Y is a cone point, we take p to be this intersection point, and if not θ u Y X θ w Y is an interior point of a saddle connection of θ u Y and we may take p to be either of its endpoints. For each component is contained in the union of the closures of such U , it follows that Π v Y pΣ w q is contained in a union of saddle connections along BΘ v Y , all of which contain Π v Y ppq, and hence is a connected union of at most two saddle connections. This completes the proof.
The final case to consider is that of spines in different directions that intersect: Lemma 3.17. There exists K 3 ą 0 such that if v, w P V with d tree pv, wq " 8 and Figure 4. The proof of Lemma 3.17 Proof. Let x 0 " θ v Y X θ w Y be the unique intersection point of the spines. Let W 0 be the smallest subgraph of θ w Y containing θ w Y X Θ v Y and let W 1 , . . . , W k be the closures of the components of θ w Y zW 0 , so that θ w Y " W 0 Y¨¨¨Y W k . See Figure 4. For 1 ď i ď k, let x i P W i be the closest (cone) point to x 0 . Then define p i to be the intersection of the geodesic rx 0 , For any x PΣ with f Y pxq P W i , where i " 1, . . . , k, the flat geodesic from Let Z P D be the closest point on BB v to BB w , thus Z P BB v lies on the unique hyperbolic geodesic that intersects BB v and BB w orthogonally. In E Z the directions αpvq and αpwq are perpendicular. Therefore the cone points of BΘ v Z that are closest to θ v Z Xθ w Z all lie in δ v Z pwq, since they must be endpoints of saddle connections along BΘ v Z that intersect θ w Z . More generally, for any Y P c v pBB w q, the directions αpvq and αpwq are nearly perpendicular and thus we have for some uniform constant K ą 0 that depends only on the length of c v pBB w q and the maximum over BB v of the length/width of any saddle connection/strip in the αpvq direction (Lemma 2.3). Now, for any x P θ w XΣ v , the point X " c v pπpxqq lies in c v pBB w q and we have that f X pxq " θ v X Xθ w X . The above equation shows there are Using the fact that the length of c v pBB w q is uniformly bounded, we see that the map f Y,X is uniformly biLipschitz and therefore that Π Combining this with the above finding that Π v Y pθ w XΣ Rv q lies within bounded distance of θ v Y X θ w Y , we finally conclude that diampΠ v Y pθ w qq is uniformly bounded.
In preparation, we first define Π v :Σ Ñ PpBΘ v q by In words, Π v pxq consists of the v-windows of x in all fibers over BB v . As before, we extend to arbitrary subsets U ĂĒ by setting Π v pU q " Π v pU XΣq. Now, for each v P V our projections are defined as the compositions Λ v " λ v˝Πv and ξ v pxq " i v˝Πv .
A useful observation is that for any two vertices v, w P V with d tree pv, wq " 1 and X PD, we have Lemma 3.18. There is a function N 1 : r0, 8q Ñ r0, 8q such that for all v P V, X P BB v , and x, y PΣ the quantities d K v px, yq and d Ξ v px, yq are at most N 1 pd v X px, yqq. Proof. For Y P BB v , let us compare the images of some subset U Ă BΘ v Y and SmearpU q " Since the boundary components of Θ v are preserved by the maps f Z,Y , the images i v pU q " i v pSmearpU qq are exactly the same. Moreover, for each x P BΘ v Y we have λ v pSmearptxuqq " λ v p x,αpvq q, and so by Proposition 3.1(2) this is a set with diameter at most K 1 . Therefore λ v pU q and λ v pSmearpU qq have Hausdorff distance at most 2K 1 . Now, let x, y PΣ and X P BB v be as in the statement. Set U " Π v X pxq Y Π v X pyq so that diampU q " d v X px, yq. Since Π v pxq " SmearpΠ v X pxqq and similarly for Π v pyq, we see that Since λ v is coarsely K 1 -Lipschitz by Proposition 3.11, the preceding paragraph and the triangle inequality shows that Similarly, by Corollary 3.9 we have that Setting N 1 ptq " maxtN ptq`1, K 1 t`3K 1 u completes the proof.
Proposition 3.19. There exists K 4 ą 0 so that for any v P V: (1) Λ v and ξ v are K 4 -coarsely Lipschitz; (2) For any w P V, we have (a) diam pΛ v pΘ w qq ă K 4 , unless d tree pv, wq ď 1; Proof. For part (1), we first observe that by [DDLS21, Lemma 3.5], there exists an R ą 0 so that any pair of points x, y PΣ may be connected by a path of length at most Rdpx, yq that is a concatenation of at most Rdpx, yq`1 pieces, each of which is either a saddle connection of length at most R in a vertical vertical fiber, or a horizontal geodesic segment inĒ. By the triangle inequality, it thus suffices to assume that x and y are the endpoints of either a horizontal geodesic or a vertical saddle connection of length at most R. Appealing to Lemma 3.18, it further suffices to show that d v X px, yq is linearly bounded bydpx, yq for some X P BB v . Lemmas 3.11 and 3.12(1) handle the horizontal segment case, since we are free to subdivide such a path into rdpx, yqs segments of length at most 1.
For the vertical segment case we assume x and y lie in the same fiber E Y and differ by a saddle connection δ of length at most R. Let θ w Y , where w P V, be the spine containing δ. The fact that δ is bounded means that Y " πpxq " πpyq is bounded distance from the horocycle BB w . Let X " c w pY q P BB w and let δ 1 " f X,Y pδq be the saddle connection in θ w X connecting x 1 " f X,Y pxq and y 1 " f X,Y pyq. By the triangle inequality and the first part above about bounded length horizontal segments, it suffices to work with the points x 1 , y 1 P θ w . There are three cases to consider: Firstly, if v " w, then x 1 , y 1 P θ v X so that Π v X px 1 q and Π v X py 1 q choose the closest cone points in BΘ v X to x 1 and y 1 , respectively. Since x 1 and y 1 are close, so are Π v X px 1 q and Π v X py 1 q. Secondly, if d tree pv, wq ą 1, then Corollaries 3.15 and 3.16, and Lemma 3.17 give a uniform bound on d v Z px 1 , y 1 q ď diampΠ v Z pθ w qq for any point Z P c v pBB w q. Finally, if d tree pv, wq " 1 then θ w X and θ v X are adjacent non-crossing spines in E X . Since θ w X is totally geodesic, if follows that Π v X px 1 q and Π v X py 1 q are either equal or connected by a single edge of BΘ v X . But this saddle connection has uniformly bounded length, since X P BB v , which completes the proof of (1).
For (2), first recall that strips/saddle connections in the αpwq direction have uniformly bounded width/length over BB w (Lemma 2.3). Therefore Θ w XΣ is contained in a bounded neighborhood of θ w XΣ. By part (1) it thus suffices to bound diampΛ v pθ w qq and diampξ v pθ w qq. When d tree pv, wq ě 2, Corollaries 3.15 and 3.16, and Lemma 3.17, imply that there exists X P BB v so that Π v X pθ w q has bounded diameter in Θ v X . Appealing to Lemma 3.18 now bounds diampΛ v pθ w qq and diampξ v pθ w qq in these cases. For the remaining case d tree pv, wq " 1 of 2(b), we note that ξ v pθ w q is a single point by (8), and thus 2(b) follows.

Hierarchical hyperbolicity of Γ
In this section we complete the proof that Γ is hierarchically hyperbolic. We will use a criterion from [BHMS20], which we now briefly discuss. For further information and heuristic discussion of this approach to hierarchical hyperbolicity, we refer the reader to [BHMS20, §1.5, "User's guide and a simple example"].
Consider a simplicial complex X and a graph W whose vertex set is the set of maximal simplices of X . The pair pX , Wq is called a combinatorial HHS if it satisfies the requirements listed in Definition 4.8 below, and [BHMS20, Theorem 1.18] guarantees that in this case W is an HHS. The main requirement is along the lines of: X is hyperbolic, and links of simplices of X are also hyperbolic. However, this is rarely the case because co-dimension-1 faces of maximal simplices have discrete links. To rectify this, additional edges (coming from W) should be added to X and its links as detailed in Definition 4.2. In our case, after adding these edges, X will be quasi-isometric toÊ, and each other link will be quasi-isometric to either a point or to one of the spaces K v or Ξ v introduced in §3.
There are two natural situations where such pairs arise that the reader might want to keep in mind. First, consider a group H acting on a simplicial complex X so that there is one orbit of maximal simplices, and those have trivial stabilizers. In this case, we take W to be (a graph isomorphic to) a Cayley graph of H. (More generally, if the action is cocompact with finite stabilizers of maximal simplices, then the appropriate W is quasi-isometric to a Cayley graph.) For the second situation, X is the curve graph of a surface; then maximal simplices are pants decompositions of the surface and W can be taken to be the pants graph. We will use this as a working example below, when we get into the details.
Most of the work carried out in §3 will be used (as a black-box) to prove that, roughly, links are quasi-isometrically embedded in a space obtained by removing all the "obvious" vertices that provide shortcuts between vertices of the link. This can be seen as an analogue of Bowditch's fineness condition in the context of relative hyperbolicity.
This section is organized as follows. In §4.1 we list all the relevant definitions and results from [BHMS20], and we illustrate them using pants graphs. In §4.2 we construct the relevant combinatorial HHS for our purposes. In §4.3 we analyze all the various links and related combinatorial objects; we note that most of the work done in §3 is used here to prove Lemma 4.22. At that point, essentially only one property of combinatorial HHSs will be left to be checked, and we do so in §4.4.

Basic definitions.
We start by recalling some basic combinatorial definitions and constructions. Let X be a flag simplicial complex.
Definition 4.1 (Join, link, star). Given disjoint simplices ∆, ∆ 1 of X, the join is denoted ∆ ‹ ∆ 1 and is the simplex spanned by ∆ p0q Y ∆ 1p0q , if it exists. More generally, if K, L are disjoint induced subcomplexes of X such that every vertex of K is adjacent to every vertex of L, then the join K ‹ L is the induced subcomplex with vertex set K p0q Y L p0q .
For each simplex ∆, the link Lkp∆q is the union of all simplices ∆ 1 of X such that ∆ 1 X ∆ " H and ∆ 1 ‹ ∆ is a simplex of X. The star of ∆ is Starp∆q " Lkp∆q ‹ ∆, i.e. the union of all simplices of X that contain ∆.
We emphasize that H is a simplex of X, whose link is all of X and whose star is all of X.
Definition 4.2 (X-graph, W -augmented dual complex). An X-graph is any graph W whose vertex set is the set of maximal simplices of X (those not contained in any larger simplex).
For a flag complex X and an X-graph W , the W -augmented dual graph X`W is the graph defined as follows: ‚ the 0-skeleton of X`W is X p0q ; ‚ if v, w P X p0q are adjacent in X, then they are adjacent in X`W ; ‚ if two vertices in W are adjacent, then we consider σ, ρ, the associated maximal simplices of X, and in X`W we connect each vertex of σ to each vertex of ρ. We equip W with the usual path-metric, in which each edge has unit length, and do the same for X`W . Observe that the 1-skeleton of X is a subgraph X p1q Ă X`W .
We provide a running example to illustrate the various definitions in a familiar situation. This example will not be used in the sequel.
Example 4.3. If X is the curve complex of the surface S, then an example of the an X-graph, W , is the pants graph, since a maximal simplex is precisely a pants decomposition. The W -augmented dual graph can be thought of as adding to the curve graph, X p0q an edge between any two curves that fill a one-holed torus or fourholed sphere and intersect once or twice, respectively: indeed, these subsurfaces are precisely those where an elementary move happens as in the definition of adjacency in the pants graph.
Definition 4.4 (Equivalent simplices, saturation). For ∆, ∆ 1 simplices of X, we write ∆ " ∆ 1 to mean Lkp∆q " Lkp∆ 1 q. We denote by r∆s the equivalence class of ∆. Let Satp∆q denote the set of vertices v P X for which there exists a simplex ∆ 1 of X such that v P ∆ 1 and ∆ 1 " ∆, i.e.
Satp∆q "¨ď We denote by S the set of "-classes of non-maximal simplices in X.
Definition 4.5 (Complement, link subgraph). Let W be an X-graph. For each simplex ∆ of X, let Y ∆ be the subgraph of X`W induced by the set of vertices pX`W q p0q´S atp∆q.
Let Cp∆q be the full subgraph of Y ∆ spanned by Lkp∆q p0q . Note that Cp∆q " Cp∆ 1 q whenever ∆ " ∆ 1 . (We emphasize that we are taking links in X, not in X`W , and then considering the subgraphs of Y ∆ induced by those links.) We now pause and continue with the illustrative example.
Example 4.6. Let X and W be as in Example 4.3. A simplex ∆ is a multicurve which determines two (open) subsurfaces U " U p∆q, U 1 " U 1 p∆q Ă S, where U is the union of the complementary components of the multicurve that are not a pair of pants, and U 1 " S´U . Note that BU Ă ∆ is a submulticurve and that ∆´BU is a pants decomposition of U 1 . A simplex ∆ 1 is equivalent to ∆ if it defines the same subsurfaces. Thus Satp∆q consists of BU p∆q together with all essential curves in U 1 p∆q, while Cp∆q is the join of graphs quasi-isometric to curve graphs of the components of U p∆q. For components of U p∆q which are one-holed tori or four-holed spheres, the corresponding subgraphs are isometric to their curve graphs (since the extra edges in X`W precisely give edges for these curve graphs).
We note that if ∆ 1 Ă ∆, then r∆s Ď r∆ 1 s. Also, for Example 4.3,4.6, r∆s Ď r∆ 1 s if and only if U p∆q Ă U p∆ 1 q.
Finally, we are ready for the main definition: Definition 4.8 (Combinatorial HHS). A combinatorial HHS pX, W q consists of a flag simplicial complex X and an X-graph W satisfying the following conditions for some n P N and δ ě 1: (1) any chain r∆ 1 s Ĺ r∆ 2 s Ĺ . . . has length at most n; (2) for each non-maximal simplex ∆, the subgraph Cp∆q is δ-hyperbolic and pδ, δq-quasi-isometrically embedded in Y ∆ ; (3) Whenever ∆ and ∆ 1 are non-maximal simplices for which there exists a nonmaximal simplex Γ such that rΓs Ď r∆s, rΓs Ď r∆ 1 s, and diampCpΓqq ě δ, then there exists a simplex Π in the link of ∆ 1 such that r∆ 1 ‹ Πs Ď r∆s and all rΓs as above satisfy rΓs Ď r∆ 1 ‹ Πs; (4) if v, w are distinct non-adjacent vertices of Lkp∆q, for some simplex ∆ of X, contained in W -adjacent maximal simplices, then they are contained in W -adjacent simplices of the form ∆ ‹ ∆ 1 .
We will see below that combinatorial HHS give HHSs. The reader not interested in the explicit description of the HHS structure can skip the following two definitions.
Definition 4.9 (Orthogonality, transversality). Let X be a simplicial complex. Let ∆, ∆ 1 be non-maximal simplices of X. Then we write r∆sKr∆ 1 s if Lkp∆ 1 q Ď LkpLkp∆qq. If r∆s and r∆ 1 s are not K-related or Ď-related, we write r∆s&r∆ 1 s. Definition 4.10 (Projections). Let pX, W q be a combinatorial HHS.
Fix r∆s P S and define a map π r∆s : W Ñ PpCpr∆sqq as follows. First let p : Y ∆ Ñ PpCpr∆sqq be the coarse closest-point projection, i.e.
Suppose that w P W p0q , so w corresponds to a unique simplex ∆ w of X. Define π r∆s pwq " pp∆ w X Y ∆ q.
We have thus defined π r∆s : W p0q Ñ PpCpr∆sqq. If v, w P W p0q are joined by an edge e of W , then ∆ v , ∆ w are joined by edges in X`W , and we let π r∆s peq " π r∆s pvq Y π r∆s pwq. Now let r∆s, r∆ 1 s P S satisfy r∆s&r∆ 1 s or r∆ 1 s Ĺ r∆s. Let ρ r∆ 1 s r∆s " ppSatp∆ 1 q X Y ∆ q.
Let r∆s Ĺ r∆ 1 s. Let ρ r∆ 1 s r∆s : Cpr∆ 1 sq Ñ Cpr∆sq be the restriction of p to Cpr∆ 1 sq X Y ∆ , and H otherwise.
The next theorem from [BHMS20] provides the criteria we will use to prove that Γ is a hierarchically hyperbolic group.
Given a combinatorial HHS pX, W q, we denote S W the set as in Definition 4.4, endowed with nesting and orthogonality relations as in Definitions 4.7 and 4.9. Also, we associated to S W the hyperbolic spaces as in Definition 4.8, and define projections as in Definition 4.10.
Theorem 4.11. [BHMS20, Theorem 1.18, Remark 1.19] Let pX, W q be a combinatorial HHS. Then pW, S W q is a hierarchically hyperbolic space.
Moreover, if a group G acts by simplicial automorphisms on X with finitely many orbits of links of simplices, and the resulting G-action on maximal simplices extends to a metrically proper cobounded action on W , then G acts metrically properly and coboundedly by HHS automorphisms on pW, S W q, and is therefore a hierarchically hyperbolic group.

4.2.
Combinatorial HHS structure. We now define a flag simplicial complex Given a vertex s P K, let vpsq P V be the unique vertex with s P K v . We also write αpsq " αpvpsqq.
There are 3 types of edges (see Figure 5): (1) v, w P V are connected by an edge if and only if d tree pv, wq " 1.
(2) s, t P K are connected by an edge if and only if d tree pvpsq, vptqq " 1.
(3) s P K and w P V are connected by an edge if and only if d tree pvpsq, wq ď 1. We declare X to be the flag simplicial complex with the 1-skeleton defined above. The map K Ñ V given by s Þ Ñ vpsq and the identity V Ñ V extends to a surjective simplicial map We note that we may view the union Ů T α on the right as a subgraph of X p1q , making Z a retraction.
For any vertex v in any tree T α , Z´1pvq is the join of tvu and the set K v : For any pair of adjacent vertices v, w P T α (so d tree pv, wq " 1), the preimage of the edge rv, ws Ă T α is also a join: Lemma 4.12. The maximal simplices of X are exactly the 3-simplices with vertex set ts, vpsq, t, vptqu where s, t P K and d tree pvpsq, vptqq " 1. In this case, we say that ps, tq defines a maximal simplex, denoted σps, tq.
Proof. Because the map Z is simplicial, any simplex of X is contained in Z´1pvq or Z´1prv, wsq for some vertex v in some T α or some edge rv, ws in some T α . The lemma thus follows from (9) and (10).
Given a vertex s P K, recall from Lemma 3.5 that M psq " pλ vpsq q´1pN K1 psqq Ă Θ vpsq , for K 1 as in Proposition 3.1 (and Lemma 3.5). Given a pair of vertices ps, tq in K that define a maximal simplex σps, tq, we will write M ps, tq " M psq X M ptq.
Lemma 4.13. There exists R ą 0 with the following properties.
(1) For any pair of adjacent vertices s, t P K (i.e., defining a maximal simplex σps, tq), M ps, tq is a non-empty subset of diameter at most R.
(4) The collection of all M ps, tq is R-dense inĒ.
Proof. Item (1) follows from Proposition 3.1(3). More precisely, the fact that M ps, tq is non-empty follows from K 1 -coarse-surjectivity of λ vpsqˆλvptq , while boundedness follows from the fact that said map is a quasi-isometry.
In order to show item (2), notice that That is, every point of Θ v is also in Θ w for some w adjacent to v. In view of this, we conclude by noticing that if x P Θ v X Θ w , then x P M pλ v pxq, λ w pxqq. Item (3) follows similarly. Finally, item (4) follows from item (2) and the fact that the collection of all Θ v is coarsely dense inĒ.
Next we define a graph W whose vertex set is the set of maximal simplices of X . We would like to just connect maximal simplices when the corresponding subsets M ps, tq are close inĒ (first bullet below); however, in order to arrange item (4) of the definition of combinatorial HHS (and only for that reason) we need different closeness constants for different situations. We fix R as in Lemma 4.13, and moreover we require R ą K 2 1`K1 , for K 1 as in Proposition 3.1 and Lemma 3.5. Given maximal simplices σps 1 , t 1 q and σps 2 , t 2 q, we declare them to be connected by an edge in W if one of the following holds: ‚dpM ps 1 , t 1 q, M ps 2 , t 2 qq ď 10R ‚ s 1 " s 2 anddpM pt 1 q, M pt 2 qq ď 10R Here the thed-distances are the infimal distances between the sets inĒ (as opposed to the diameter of the union). Note that since M ps, tq " M pt, sq, the second case also implicitly describes a "symmetric case" with s i and t i interchanged.
The following is immediate from Lemma 3.6, setting R 1 " maxt10R, N K p10Rqu.
Lemma 4.14. There exists R 1 ě 10R so then the following holds. If s, t 1 , t 2 P K are vertices with s connected to both t i in X anddpM pt 1 q, M pt 2 qq ď 10R, then dpM ps, t 1 q, M ps, t 2 qq ď R 1 . In particular, whenever σps 1 , t 1 q and σps 2 , t 2 q are connected in W, we havedpM ps 1 , t 1 q, M ps 2 , t 2 qq ď R 1 .
Lemma 4.15. W is quasi-isometric toĒ, by mapping each σps, tq to (any point in) M ps, tq. Moreover, the extension group Γ acts by simplicial automorphisms on X , induced by the existing action on V Ď X p0q and the action on K Ď X p0q as in Proposition 3.1(4). The resulting action on maximal simplices extends to a metrically proper cobounded action on W.
Proof. In view of Lemma 4.14, the first part follows by combining Lemma 4.13(4) and Proposition 2.4 (applied to any choice of a point in each M ps, tq).
It is immediate to check that the Γ-action defined on the 0-skeleton of X extends to an action on X . That the resulting action on maximal simplices of X (that is, the 0-skeleton of W) extends to an action on W follows from the equivariance property in Proposition 3.1(4) and the definitions of the sets M psq and M ps, tq.
Moreover, the quasi-isometry W ÑĒ described in the statement is Γ-equivariant, so that the action of Γ on W is metrically proper and cobounded since the action of Γ onĒ has these properties.
The goal for the remainder of this section is to prove the following.  Proof. We define a map Z 1 : X`W ÑÊ that extends the (restricted) simplicial map Z : X p1q Ñ Ů T α already constructed above. To do that, we must extend over each edge e " rx, ys of X`W coming from the edge of W connecting σps 1 , t 1 q and σps 2 , t 2 q. Since Zpxq, Zpyq P V, anddpM ps 1 , t 1 q, M ps 2 , t 2 qq ď R 1 (for R 1 as in Lemma 4.14), we see that dÊpv, wq ď R 1 . We can then define Z 1 on e to be a constant speed parameterization of a uniformly bounded length path from Zpxq to Zpyq. It follows that Z 1 is Lipschitz.
The union of the trees Ů T α is R 0 -dense for some R 0 ą 0 by [DDLS21, Lemma 3.6], so it suffices to find a one-sided inverse to Z 1 , from Ů T α to X`W , and show that with respect to the subspace metric fromÊ, it is coarsely Lipschitz. As already noted, Z restricts to a retraction of X p1q onto Ů T α Ă X p1q Ă X`W , which is thus the required one-sided inverse. All that remains is to show that it is coarsely Lipschitz.
According to [DDLS21,Lemma 3.8], any v P T α , w P T β are connected by a combinatorial path of length comparable todpv, wq. Such a path is the concatenation of horizontal jumps, each of which is theP -image inÊ of a geodesic inD z , for some z PΣ, that connects two components of BD z and whose interior is disjoint from BD z . From that same lemma, we may assume each horizontal jump has length uniformly bounded above and below, and thus has total number of jumps bounded in terms ofdpv, wq. Therefore, we can reduce to the case that v, w are joined by a single horizontal jump of bounded length. Such a horizontal jump can also be regarded as a path inĒ connecting Θ v to Θ w . Hence, in view of Lemma 4.13(2), there are M ps 1 , t 1 q Ď Θ v and M ps 2 , t 2 q Ď Θ w within uniformly bounded distance of each other inĒ. Lemma 4.15 implies that there exists a path in W of uniformly bounded length from σps 1 , t 1 q to σps 2 , t 2 q, which can be easily turned into a path of uniformly bounded length from v to w in X`W , as required.
There is an important type of 1-dimensional simplex, which we call a Ξ-type simplex, due to the following lemma. See Figure 6. Given w P V, set Lemma 4.18 (Ξ-type simplex). Let ∆ be a 1-simplex of X with vertices s, vpsq, for s P K. Then Lkp∆q " Lk Ξ pvpsqq and Satp∆q " tvpsqu Y K vpsq .
Moreover, Cp∆q is quasi-isometric to Ξ vpsq , via a quasi-isometry which is the identity on V X Cp∆q and maps t to vptq for t P K X Cp∆q. Proof. It is clear from the definitions that the link of ∆ is as described. Also, any simplex with vertex set of the form tvpsq, tu for some t P K vpsq has the same link as ∆. Therefore, to prove that the saturation is as described we are left to show if a simplex ∆ 1 has the same link as ∆, then its vertex set is contained in the set we described. If w P V is a vertex of ∆ 1 , then d tree pv, wq " 1 for all neighbors v of vpsq in T αpvq . This implies w " vpsq. Similarly, if t P K is a vertex of ∆ 1 , then vptq " vpsq, and we are done.
Let us show that the map given in the statement is coarsely Lipschitz. To do so, it suffices to consider w ‰ w 1 with d tree pw, vpsqq " d tree pw 1 , vpsqq " 1 and connected by an edge in X`W , and show that they are connected by a bounded-length path in Ξ v . We argue below thatdpΘ w X Θ vpsq , Θ w 1 X Θ vpsq q ď R 1 . Once we do that, the existence of the required bounded-length path follows directly from Lemma 3.8.
Let us now prove the desired inequality. Notice that w, w 1 cannot be connected by an edge of X since they are both distance 1 from vpsq in the tree T αpvq . Hence, w and w 1 are contained respectively in maximal simplices σpt, uq and σpt 1 , u 1 q connected by an edge in W. Say, up to swapping t with u and/or t 1 with u 1 , that vptq " w and vpt 1 q " w 1 . In either case of the definition of the edges of W we havē dpM ptq, M pt 1 qq ď 10R. Since M pt, sq Ď Θ w X Θ vpsq and M pt 1 , sq Ď Θ w 1 X Θ vpsq , using Lemma 4.14 we get as we wanted. Conversely, if w, w 1 as above are joined by an edge in Ξ vpsq we will now show that they are also connected by an edge in Cp∆q. By definition of Ξ vpsq , we have Θ w X Θ w 1 ‰ H. By Lemma 4.13(2), There exists t, u, t 1 , u 1 with vptq " w and vpt 1 q " w 1 , so that M pt, uq X M pt 1 , u 1 q ‰ H. In particular,dpM pt, uq, M pt 1 , u 1 qq ď 10R. This says that σpt, uq and σpt 1 , u 1 q are connected in W, and hence that w and w 1 are connected in X`W , as required.
There is also an important type of 2-dimensional simplex, which we call a K-type simplex, due to the next lemma. See Figure 7.
Lemma 4.19 (K-type simplex). Let ∆ be a 2-simplex of X with vertices s, vpsq, w, for s P K and w P V with d tree pw, vpsqq " 1. Then Lkp∆q " K w and Satp∆q " twu Y Lk Ξ pwq p0q .
Moreover, Cp∆q is quasi-isometric to K w , the quasi-isometry being the identity at the level of vertices. Proof. It is clear from the definitions that the link of ∆ is as described. Also, any simplex with vertex set of the form tt, vptq, wu for some t P K with d tree pvptq, wq " 1 has the same link as ∆. Therefore, to prove that the saturation is as described we are left to show if a simplex ∆ 1 has the same link as ∆, then its vertex set is contained in the set we described. Given any vertex u P V of ∆ 1 , it has to be connected to all t P K with vptq " w, implying that either u " w or d tree pu, wq " 1, as required for vertices in V. Similarly, any vertex u P K of ∆ 1 has to be connected to all t P K with vptq " w, implying d tree pvpuq, wq " 1, and we are done.
To prove that Cp∆q is naturally quasi-isometric to K w , it suffices to show that if u, t P Cp∆q are connected by an edge in X`W , then they are uniformly close in K w and that, vice versa, if d K w pu, tq " 1, then u, t are connected by an edge in Cp∆q.
First, if u, t P Cp∆q are connected by an edge in X`W , then there exist u 1 , t 1 so thatdpM pu, t 1 q, M pu 1 , tqq ď R 1 (see Lemma 4.14). In particular,dpM puq, M ptqq ď R 1 , which in turn gives a uniform bound on the distance in the path metric of Θ w between M puq and M ptq because the metricsd and the path metric on θ w are coarsely equivalent. By Proposition 3.1(1) we must also have a uniform bound on d K w pu, tq. Now suppose that d K w pu, tq " 1. We can then deduce from Proposition 3.1(3) thatdpM ps 1 , uq, M ps 1 , tqq ď K 2 1`K1 ď 10R for any fixed s 1 P K vpsq (the proposition yields the analogous upper bound in the path metric of Θ w , which is a stronger statement). Therefore u, t are connected by an edge in X`W , whence in Cp∆q, as required.
The remaining simplices are not particularly interesting as their links are joins (or points), and hence have diameter at most 3, but we will still need to verify properties for them. We describe each type and its link in the next lemma. Recall the definition of Lk Ξ pwq in (12).
Lemma 4.20. The following is a list of all types of non-empty, nonmaximal, simplices ∆ of X that are not of Ξ-type or K-type, together with their links. In each case, the link is a nontrivial join (or a point), and Cp∆q has diameter at most 3.
In the table below the simplices ∆ have vertices u, w P V with d tree pu, wq " 1 and s, t P K with d tree pvpsq, vptqq " 1 and d tree pvpsq, uq " 1.
Proof. This is straightforward given the definition of X and we leave its verification to the reader. Referring to Figure 5, and comparing with Figures 6 and 7, may be helpful.
The next lemma collects a few additional properties we will need. By the type of a simplex in X , we mean its orbit by the action of the simplicial automorphism group of X . There are 9 types of nonempty simplices: maximal, Ξ-type, K-type, and the six types listed in Lemma 4.20.
Lemma 4.21. The following hold in X .
paq The link of a simplex with a given type cannot be strictly contained in the link of a simplex with the same type. pbq For all non-maximal simplices ∆ and ∆ 1 so that there is a simplex Γ with LkpΓq Ď Lkp∆ 1 q X Lkp∆q and diampCpΓqq ą 3, there exists a simplex Π in the link of ∆ 1 with Lkp∆ 1˚Π q Ď Lkp∆q so that for any Γ as above we have LkpΓq Ď Lkp∆ 1˚Π q.
Proof. Part paq follows directly from the descriptions of the simplices given in Lemmas 4.18, 4.19, and 4.20, and we leave this to the reader. Before we prove pbq, we suppose Lkp∆ 1 q X Lkp∆q ‰ H, and make a few observations. First, ∆ 1 and ∆ must project by Z to the same tree: Zp∆ 1 q, Zp∆q Ă T α for some α P P. Next, note that ZpLkp∆ 1 q X Lkp∆qq is contained in the intersection of the stars in T α of Zp∆ 1 q and Zp∆q. Moreover, (as in any tree) the intersection of these two stars is contained in a single edge, or else Zp∆ 1 q " Zp∆q " twu P T p0q α Ă V is a single point. In this latter case, by (9), we have Next, note that for any K-type simplex Γ " ts, vpsq, wu, LkpΓq " K w by Lemma 4.19, and if LkpΓq Ă Lkp∆ 1 q X Lkp∆q, then w is in the intersection of the stars of Zp∆ 1 q and Zp∆q. For a Ξ-type simplex Γ " ts, vpsqu, LkpΓq " Lk Ξ pvpsqq by Lemma 4.18, and together with Lemma 4.20 and the previous paragraph, we see that LkpΓq Ă Lkp∆ 1 q X Lkp∆q if and only if Zp∆ 1 q " Zp∆q " tvpsqu in T α .
With these observations in hand, we proceed to the proof of pbq, which divides into two cases.
In this situation, Zp∆ 1 q " Zp∆q " tvpsqu, and thus ∆, ∆ 1 Ă tvpsqu ‹ K vpsq . From Lemmas 4.18 and 4.20, we see that Lkp∆ 1 q X Lkp∆q must be equal to one of LkpΓq, Lkptsuq, or Lkptvpsquq. Inspection of these links shows that LkpΓq is the only link of a Ξ-type simplex contained in it. First suppose that Lkp∆ 1 q X Lkp∆q has the form LkpΓq or Lkptsuq. In this situation, we easily find Π Ă Lkp∆ 1 q so that LkpΓq " Lkp∆ 1 ‹ Πq. Furthermore, for any K-type simplex link K w in the intersection, we must have K w Ă LkpΓq " Lkp∆ 1 ‹ Πq. Therefore, the link of any Ξ-or K-type simplex contained in Lkp∆ 1 q X Lkp∆q must be contained in LkpΓq " Lkp∆ 1 ‹ Πq, as required. Now suppose instead that Lkp∆ 1 qXLkp∆q " Lkptvpsquq. By Lemma 4.20, we see that ∆ " ∆ 1 " tvpsqu. In this case, setting Π " H trivially completes the proof since then Lkp∆ 1 q X Lkp∆q " Lkp∆ 1 q " Lkp∆ 1 ‹ Πq.
Case 2. No link of a Ξ-type simplex is contained in Lkp∆ 1 q X Lkp∆q.
From the observations above, Zp∆q and Zp∆ 1 q do not consist of the same single point, and hence the stars of Zp∆q and Zp∆ 1 q intersect in either a point or an edge in T α . Since ZpLkp∆ 1 qXLkp∆qq is contained in the intersection of these stars, there are at most two K-type simplices whose links are contained in Lkp∆ 1 q X Lkp∆q. If there are two K-type simplices Γ, Γ 1 with K w " LkpΓq and K u " LkpΓ 1 q contained in Lkp∆ 1 q X Lkp∆q, then observe that By inspection of the possible links in Lemmas 4.18, 4.19, and 4.20, it must be that ∆ 1 is either tuu, twu, or tu, wu, and so setting Π to be twu, tuu, or H, respectively, we are done. On the other hand, if there is exactly one K-type simplex Γ with K w " LkpΓq Ă Lkp∆ 1 q X Lkp∆q, then again inspecting all possible situations, we can find Π Ă Lkp∆ 1 q with K w " LkpΓq " Lkp∆ 1 ‹ Πq, and again we are done with this case. This completes the proof.
Lemma 4.22. There exists L ě 1 so that for every non-maximal simplex, there is an pL, Lq-coarsely Lipschitz retraction r ∆ : Y ∆ Ñ PpCp∆qq. In particular, Cp∆q is uniformly quasi-isometrically embedded in Y ∆ .
Proof. By Lemma 4.20, we only have to consider simplices of Ξ-and K-type.
Consider ∆ " ts, vu with v " vpsq of Ξ-type first. Recall from Lemma 4.18 that Ξ v naturally includes into Cp∆q by a quasi-isometry. Here we will use make use of the map ξ v , whose relevant properties for our current purpose are stated in Proposition 3.19. For a vertex u P V Satp∆q (so, u ‰ v) we define r ∆ puq " ξ v pΘ u q. For t P K Satp∆q we define r ∆ ptq " ξ v pΘ vptq q. Notice that the sets r ∆ puq are uniformly bounded by Proposition 3.19 (and Lemma 4.18). Also, r ∆ is coarsely the identity on the vertices of Lkp∆q in V by Equation (8) and Proposition 3.19(2b). To check that r ∆ is coarsely Lipschitz it suffices to consider X`W -adjacent vertices of V. Notice that vertices w, w 1 P V that are adjacent in X`W have corresponding Θ w , Θ w 1 within 10R of each other inĒ. Indeed, Θ v and Θ w actually intersect if w, w 1 are adjacent in X , and they contain subsets M p¨q within 10R of each other if w, w 1 are contained in W-adjacent maximal simplices (this is true regardless of which case of the definition (11) for the edges of W applies). The fact that r ∆ is coarsely Lipschitz now follows from, Proposition 3.19(1), which says that ξ v is coarsely Lipschitz onĒ.
Next, consider ∆ " ts, vpsq, wu of K-type. For a vertex u P V Satp∆q (so, d tree pu, wq ě 2), define r ∆ puq " Λ w pΘ u q. For a vertex t P K Satp∆q, define r ∆ ptq " Λ w pM ptqq. Notice that, by definition of M ptq, if t P K w , then r ∆ ptq lies within Hausdorff distance K 1 of t. Also, since d tree pu, wq ě 2, Proposition 3.19(2a) ensures that the diameter of Λ w pΘ u q is bounded. Since M ptq Ď Θ vptq , we see that all the sets in the image of r ∆ are bounded, and also we see that in order to prove that r ∆ is Lipschitz it suffices to consider vertices of K. But vertices s, t P K that are adjacent in X`W have corresponding M psq, M ptq within 10R of each other inĒ, so the conclusion follows from Proposition 3.19(1), which states that Λ w is coarsely Lipschitz onĒ.

Final proof.
We now have all the tools necessary for the: Proof of Theorem 4.16. We must verify each of the conditions from Definition 4.8.
Item (1) (bound on length of Ď-chains) follows from Lemma 4.21(a), which implies that any chain Lkp∆ 1 q Ĺ . . . has length bounded by the number of possible types, which is 9.
Let us now discuss item (2) of the definition. The descriptions of the Cp∆q from Lemmas 4.17, 4.18, 4.19, 4.20 yields that all Cp∆q are hyperbolic, since each of them is either bounded or uniformly quasi-isometric to one ofÊ (which is hyperbolic by Theorem 2.1), some Ξ v (which is hyperbolic by Lemma 3.10), or R. Moreover, any Cp∆q is (uniformly) quasi-isometrically embedded in Y ∆ by Lemma 4.22.
Item (3) of the definition (common nesting) is precisely Lemma 4.21(b). Finally, we show item (4) of the definition (fullness of links), which we recall for the convenience of the reader: ‚ If v, w are distinct non-adjacent vertices of Lkp∆q, for some simplex ∆ of X , contained in W-adjacent maximal simplices, then they are contained in W-adjacent simplices of the form ∆ ‹ ∆ 1 .
It suffices to consider simplices ∆ of Ξ-and K-type. Indeed, in all other cases (see Lemma 4.20), the vertices v, w under consideration are contained in the link of a simplex ∆ 1 containing ∆ where ∆ 1 is of Ξ-or K-type (as can be seen by enlarging ∆ until its link is no longer a join; v and w are not X -adjacent so they are contained in the same "side" of any join structure). Hence, once we deal with those cases, we know that there are suitable maximal simplices containing the larger simplex, whence ∆.
Consider first a simplex ∆ of K-type with vertices s, vpsq, w. Consider distinct vertices t 1 , t 2 (necessarily in K) of Lkp∆q, and suppose that there are vertices s 1 , s 2 P K so that the maximal simplices σps 1 , t 1 q and σps 2 , t 2 q are connected in W. There are two possibilities: ‚dpM ps 1 , t 1 q, M ps 2 , t 2 qq ď 10R. In this case, we havedpM pt 1 q, M pt 2 qq ď 10R. In particular, in view of the second bullet in the definition of the edges of W, we have that t 1 , t 2 are contained, respectively, in the W-connected maximal simplices ∆˚t 1 " σps, t 1 q and ∆˚t 2 " σps, t 2 q. ‚ s 1 " s 2 anddpM pt 1 q, M pt 2 qq ď 10R (notice that t 1 ‰ t 2 so that the "symmetric" case cannot occur). Again, we reach the same conclusion as above.
We can now consider a simplex ∆ of Ξ-type with vertices s, vpsq. Consider vertices x 1 , x 2 of Lkp∆q that are not X -adjacent but are contained in W-adjacent maximal simplices. Furthermore, we can assume that x 1 , x 2 are not in the link of a simplex of K-type (the case we just dealt with) which contains ∆, since in that case we already know that there are suitable maximal simplices containing the larger simplex, whence ∆. Then, using the structure of Lkp∆q, we see that there are vertices s i , t i P K so that: ‚ x i P tt i , vpt i qu, ‚ t i and vpt i q all belong to Lkp∆q, and ‚ σps 1 , t 1 q and σps 2 , t 2 q are connected in W.
We now also have to check the existence of an action of Γ with the required properties. The action is constructed in Lemma 4.15, where all properties are checked except finiteness of the number of orbits of links of X . The finitely many possible types of links are listed in Lemmas 4.17 -4.20, and for each type of simplex there are only finitely many orbits, so we are done.

Quasi-isometric rigidity
In this section, using the HHS structure, we prove a strong form of rigidity for the group Γ and the model spaceĒ. Recall thatĒ is defined via a particular trun-cationD of the Teichmüller disk D obtained by removing 1-separated horoballs. We say that such a truncationĒ is an allowable truncation of E if Γ acts by isometries on it with cocompact quotient. Write IsompΩq and QIpΩq for the isometry group and quasi-isometry group, respectively, of a metric space Ω. ForĒ, we write Isom fib pĒq ď IsompĒq for the subgroup of isometries that map fibers to fibers.
The proof is divided up into several steps which we outline here before getting into the details. The first step is to use the HHS structure to identify certain quasiflats inĒ, and prove that they are coarsely preserved by a quasi-isometry. The maximal quasi-flats are encoded by the strip bundles inĒ, and using the preservation of quasi-flats, we show that a quasi-isometry further preserves strip bundles, and even sends all strip bundles for strips in any fixed direction to strip bundles in some other fixed direction. From there we deduce that a quasi-isometry sends fibers E X within a bounded distance of some other fibers E Y , and in fact induces a quasi-isometry between the fibers. Fixing attention on E 0 and further appealing to the structure of strip bundles, we show that a self quasi-isometry ofĒ induces a special type of quasi-isometry from E 0 to itself sending strips in a fixed direction within a uniformly bounded distance of strips in some other fixed direction. This quasi-isometry is promoted to a piecewise affine biLipschitz map from E 0 to itself, which we then show is in fact affine. This produces a homomorphism to the full affine group of E 0 , QIpĒq Ñ AffpE 0 q. Given an affine homeomorphism of E 0 , we construct an explicit fiber preserving isometry associated to it, which via the inclusions Isom fib pĒq Ñ QIpĒq serves as a one-sided inverse. Finally, we prove that the homomorphism QIpĒq Ñ AffpE 0 q is injective, hence the homomorphisms Isom fib pĒq Ñ IsompĒq Ñ QIpĒq Ñ AffpE 0 q are all isomorphisms. The fact that Γ has finite index in Isom fib pĒq, and hence in IsompĒq, is straightforward using the cocompactness of the action of Γ and the singular structure. 5.1. HHS structure and quasi-flats. Denote by S 0 the set Vˆt0, 1u. We denote the element pv, 0q by v qt (for "quasi-tree") and pv, 1q by v ql (for "quasi-line").
We denote by F the set of all strip bundles ofĒ, that is, subbundles with fiber a strip and base the horocycle corresponding to the direction of the strip. (Roughly, these are the flats of the peripheral graph manifolds.) Proposition 5.1 (Properties of the HHS structure). The HHS structure pĒ, Sq onĒ coming from Theorem 4.16 has the following properties, for some K ě 1.
(1) The set of non-Ď-maximal Y P S with diampCpY qq ě 4 is in bijection with S 0 . Under said bijection: (2) Cpv qt q is pK, Kq-quasi-isometric to a quasi-tree with at least two points at infinity, and Cpv ql q is pK, Kq-quasi-isometric to a line; (3) For all v P V, we have v qt Kv ql ; (4) For all adjacent v, w P V, we have w ql Kv ql and w ql Ď v qt ; (5) All pairs of elements of S 0 that do not fall into the aforementioned cases are transverse; (6) For each adjacent v, w P V there is F P F so π v ql pF q and π w ql pF q are K-coarsely dense, and π Y pF q has diameter at most K for all Y ‰ v ql , w ql .
Proof. The second paragraph of the proof of Theorem 4.16 implies Cpv qt q is quasiisometric to the quasi-tree Ξ v (with at least two points at infinity by Lemma 3.10) and Cpv ql q is quasi-isometric to the quasi-line K v , and that these are the only nonmaximal elements of diameter at least 4. This proves (1)  The required strip bundle is the intersection Θ v X Θ w , which under the quasiisometry of Lemma 4.15 corresponds to the set of all maximal simplices of W of the form σps, tq for s P K v , t P K w . In view of the description of the π Y from Definition 4.10, the coarse density claim follows since the union of the simplices described above contains the links of the simplices corresponding to v ql and w ql , which are K v and K w .
Regarding the boundedness claim, it can be checked case-by-case that the set of simplices described above gives a bounded set of Y ∆ for r∆s ‰ v ql , w ql (for example, note that said set is bounded if the saturation of ∆ does not intersect K v Y K w , or if it does not contain v or w). This implies boundedness of the projections since the projections are coarsely Lipschitz; this follows from Theorem 4.16 since the projection maps of an HHS are required to be coarsely Lipschitz.
From now on we identify S 0 with the set of all Y P S with diampCpY qq ě 4 as in Proposition 5.1. Notice that the maximal number of pairwise orthogonal elements of S 0 is 2. Therefore, a complete support set as in [BHS21, Definition 5.1] is just a pair of orthogonal elements of S 0 .
Let H be the set of pairs pY, pq where Y P S and p P BCpY q with Y " v ql for some v P V. We say that two such pairs pY, pq and pW, qq are orthogonal if Y and W are. Any element σ " pY, pq P H comes with a quasi-geodesic ray h σ inĒ, as in [BHS21,Definition 5.3], so that π Y˝hσ is a quasi-geodesic in CpY q and π W ph σ q is bounded for all W ‰ Y .
We recall that given subsets A and B of a metric space X, we say that the subset C of X is the coarse intersection of A and B if for every sufficiently large R we have that N R pAq X N R pBq lies within finite Hausdorff distance of C. If the coarse intersection of two subsets exists, then it is well-defined up to finite Hausdorff distance.
Proof. Let H 1 be the set of pairs pY, pq where Y P S and p P BCpY q, without the restriction on Y .
The lemma with H 1 replacing H would follow directly from [BHS21, Theorem 5.7], except that the HHS structure onĒ does not satisfy one of the 3 required assumptions, namely Assumption 2 (while it does satisfy Assumption 1 by parts (1) and (2) of Proposition 5.1, and it also satisfies Assumption 3 since there are no 3 pairwise orthogonal elements of S 0 , by parts (3)-(5) of Proposition 5.1).
Inspecting the proof of [BHS21, Theorem 5.7], we see that Assumption 2 is used in two places.
The first one is to define the map φ H on a certain pair σ " pY, pq P H 1 . The argument applies verbatim if Y satisfies Assumption 2, that is, if and only if Y is the intersection of 2 complete support sets. This is the case if Y " v ql for some v P H, that is, if σ P H. Therefore, one can use that argument to define a map φ H : H Ñ H 1 . What is more, the image of φ H needs to be contained in H. This can be seen from the fact that h φ H pσq for σ P H arises as a coarse intersection of standard orthants, which are, essentially, products of rays h σ , see [BHS21,Definition 4.1] for the precise definition. Notice that [BHS21,Lemma 4.11] says, roughly, that coarse intersections of standard orthants are the expected sub-products. Hence, the failure of Assumption 2 for Y " v qt implies that h pY,pq cannot be a coarse intersection of standard orthants, and therefore φ H pσq for σ P H also needs to lie in H.
The second place where Assumption 2 is mentioned in [BHS21, Theorem 5.7] is the proof that φ H preserves orthogonality. There the assumption is used to say that certain quasi-geodesic rays are of the form h σ . Such quasi-geodesic rays arise as coarse intersections of standard orthants, so, as mentioned above, they need to be of the form h σ for σ P H, hence Assumption 2 is not actually needed there.
Thus, the arguments in the proof of [BHS21, Theorem 5.7] give the lemma.
Lemma 5.3. For every K there exists C so that the following holds. Let φ :Ē Ñ E be a pK, Kq-quasi-isometry. Then there is a bijection φ F : F Ñ F so that d Haus pφpF q, φ F pF qq ď C for all F P F.
Proof. Let p˘be the two points at infinity of Cpv ql q for some v P V. We claim that there exists w P V so that, for q˘the points at infinity of Cpw ql q, we have φ H ppv ql , p˘qq " pw ql , q˘q, up to relabeling. We use that φ H preserves orthogonality to show this. Let u 1 , u 2 P V be distinct and adjacent to v, and let r1 , r2 be the points at infinity of Cpu ql 1 q, Cpu ql 2 q. Then pv ql , p˘q are the only elements of H that are orthogonal to all the pu ql i , rȋ q. Since φ H preserves orthogonality, we see that φ H ppv ql , p˘qq are both orthogonal to the same 4 distinct elements of H with the property that no pair of them is orthogonal. This is easily seen to imply that φ H ppv ql , p˘qq must be of the form pw, q˘q, since said 4 elements need to be associated to at least 2 distinct vertices of V. This shows the claim.
In view of the claim, we see that [BHS21, Lemma 5.9] applies. (We note that Assumption 2 in said Lemma is only needed to have the map from [BHS21, Theorem 5.7], but our map from Lemma 5.2 has the same defining properties, just with a smaller domain and range.) The standard flats in [BHS21, Lemma 5.9] coarsely coincide with the elements of F in view of Proposition 5.1(6) (compare with [BHS21, Definition 4.1]) so the lemma follows.
Denote by S the collection of all strips in E 0 , and for A P S denote by αpAq P P the direction of A. Similarly, for F P F we denote αpF q P P the direction of the strip defining F .
Proposition 5.4. Given K ě 1, there exists C ě 0 so that if φ :Ē ÑĒ is a pK, Kq-quasi-isometry, then for all X PD, there exists Y PD so that the Hausdorff distance between φpE X q and E Y is at most C. In particular, d Haus pE 0 , φpE 0 qq ă 8. Moreover, there are bijections φ P : P Ñ P and φ S : S Ñ S so that: (1) d Haus pφpBB α q, BB φ P pαq q ď C for each α P P.
(2) αpφ S pAqq " φ P pαpAqq and d Haus pφpAq, φ S pAqq ă 8 for all A P S, Proof. First, note that fibers are quantitatively coarse intersections of the sets BB α , in the sense that exists a function f : R Ñ R and t 0 ě 0 such that ‚ for any X PD and any t ě t 0 there are two distinct BB α whose tneighborhoods intersect in a set within Hausdorff distance f ptq of E X ; ‚ for t ě t 0 , if the t-neighborhoods of two distinct BB α intersect, then this intersection lies within Hausdorff distance f ptq of a fiber.
This follows via the bundle-map π :Ē ÑD and the corresponding relationship between neighborhoods of distinct horocycles BB α inD. We next make three preliminary observations. Firstly, for each α P P the set BB α is the union of all F P F with αpF q " α. Secondly, if F 1 , F 2 P F have αpF 1 q ‰ αpF 2 q then the coarse intersection of F 1 and F 2 is bounded. Indeed, F i is contained in BB αpFiq , and the coarse intersection of these BB αpFiq is some (or really, any) fiber E X . Since the coarse intersection of F i with E X is a strip in the corresponding direction, and strips in different directions have bounded coarse intersection, the claim follows. Thirdly, observe that F 1 , F 2 P F have αpF 1 q " αpF 2 q if and only if there is a chain of elements in F from F 1 to F 2 so that consecutive elements have unbounded coarse intersection. The "if" part follows from the previous observation, while the "only if" follows from the fact that elements of F corresponding to adjacent edges of some T α have unbounded coarse intersection.
In view of all this and Lemma 5.3, we see that for each α there exists a (necessarily unique) φ P pαq P P so that φpBB α q and BB φ P pαq have finite Hausdorff distance. In fact, the distance is uniformly bounded by the constant C, depending only on K, coming from Lemma 5.3. This is how we define φ P . Now for any X PD, we may choose α 1 , α 2 P P so that the fiber E X has Hausdorff distance at most f pt 0 q from the intersection of the t 0 -neighborhoods of BB αi . Thus there is some uniform t 1 0 ě t 0 , again depending only on K, so that φpE X q has Hausdorff distance at most t 1 0 from the intersection of the t 1 0 -neighborhoods of BB φ P pαiq ; further, as mentioned above, this intersection of t 1 0 -neighborhoods has Hausdorff distance at most f pt 1 0 q to some fiber E Y , as claimed. Finally, we define φ S via the bijection F Ø S between strip bundles and strips in E 0 . That is, if A P S corresponds to F P F, then φ S pAq is the strip corresponding to φ F pF q. Since A is the coarse intersection of F with E 0 , the desired properties for φ S then follow from the facts that α φ F pF q " φ P pαpF qq and that φ F pF q lies within finite Hausdorff distance of φpF q.

From
QIpĒq to QIpE 0 q. The next step is to construct a homomorphism QIpĒq Ñ QIpE 0 q by associating a quasi-isometry of E 0 to each quasi-isometry of E (see Lemma 5.7). This step requires some preliminaries which we now explain.
To distinguish between two relevant notions of properness, we will call a map f : X Ñ Y between metric spaces topologically proper if it is continuous and preimages of compact sets are compact, and metrically proper if there exist diverging functions ρ´, ρ`: R ě0 Ñ R ě0 (which we will call properness functions) such that for all x, y P X we have ρ´pd X px, yqq ď d Y pf pxq, f pyqq ď ρ`pd X px, yqq.
(Both types of maps are just referred to as "proper" in the appropriate contexts, but neither notion implies the other.) For R ą 0 and X PD, we endow N R pE X q with the restriction of the metric of E, while E X is endowed with its path metric. Then the restriction of f X :Ē Ñ E X to N R pE X q is metrically proper. Indeed, f X is topologically proper and equivariant with respect to a group acting cocompactly. Note that the properness functions here can be taken independently of the fiber X (once we fix R) because there is also a cocompact action onD. We also note the following lemma.
Lemma 5.5. A metrically proper coarsely surjective map between geodesic metric spaces is a quasi-isometry. Moreover, the quasi-isometry constants depend only on the properness functions and the coarse surjectivity constant.
This follows from standard arguments. First, a metrically proper map from a geodesic metric space is coarsely Lipschitz (the proof involves subdividing geodesics into segments of length at most 1, each of which has bounded image). Also, coarse surjectivity allows one to construct a quasi-inverse of the map, which is furthermore metrically proper. As above, the quasi-inverse is coarsely Lipschitz, and we conclude since a coarsely Lipschitz map with a coarsely Lipschitz quasi-inverse is a quasiisometry.
Given any quasi-isometry φ :Ē ÑĒ and X PD, define ν X φ : E X Ñ E X to be ν X φ " f X˝φ | E X . In the case of the base fiber X " X 0 we denote this ν φ " ν X0 φ . When φ is understood, we also write ν X " ν X φ and ν " ν φ . Lemma 5.6. For any pK, Kq-quasi-isometry φ :Ē ÑĒ and X PD, the map ν X φ : E X Ñ E X is a pK 1 , K 1 q-quasi-isometry, where K 1 depends only on K and d Haus pE X , φpE X qq. Furthermore, for any A P S, d Haus pν X φ pAq, φ S pAqq ă 8. Proof. First note that the restriction of φ to E X is metrically proper, since the path metric on E X and the restricted metric fromĒ are coarsely equivalent (that is, the identity on E X is a metrically proper map between these metric spaces). Next let R " d Haus pE X , φpE X qq, which is finite by Proposition 5.4, and note that the restriction f X | N R pE X q : N R pE X q Ñ E X is also metrically proper. Therefore the composition ν X φ " pf X | N R pE X q q˝pφ| E X q is metrically proper and, moreover, the properness functions depend only on K, R and not on the fiber E X .
By [KL12,Theorem 3.8] and the fact that E X is uniformly quasi-isometric to H 2 , any metrically proper map of E X to itself is coarsely surjective and, moreover, the coarse surjectivity constant depends only on the properness functions. Therefore ν X φ is coarsely surjective and a uniform quasi-isometry by Lemma 5.5.
Regarding the claim about A, this follows from Proposition 5.4(2) and the fact that f X moves each point of φpAq Ď N R pE X q at most R away.
Proof. Given any quasi-isometry φ :Ē ÑĒ and x P E 0 we have dpφpxq, ν φ pxqq " dpφpxq, f 0 pφpxqq ď d Haus pφpE 0 q, E 0 q The right hand side is finite by Proposition 5.4, so the left hand side is bounded, independent of x. From this, the triangle inequality, and the uniform metric properness of the inclusion of E 0 intoĒ, it easily follows that if φ and φ 1 are bounded distance, then so are ν φ and ν φ 1 . Therefore the assignment φ Þ Ñ ν φ descends to a well-defined function A 0 : QIpĒq Ñ QIpE 0 q.
To see that A 0 is a homomorphism, suppose φ, φ 1 are pK, Kq-quasi-isometries of E. Then from the inequality above, for all x P E 0 we have dpφ 1˝φ pxq, φ 1˝ν φ pxqq ď Kdpφpxq, ν φ pxqq`K ď Kd Haus pφpE 0 q, E 0 q`K. The left-hand side is thus uniformly bounded, independent of x. From this, the triangle inequality, and Proposition 5.4, it follows that d Haus pφ 1˝φ pE 0 q, E 0 q and d Haus pφ 1˝ν φ pE 0 q, E 0 q are bounded by some constant r ą 0. Then for all x P E 0 , dpν φ 1˝φpxq, ν φ 1˝ν φ pxqq " dpf 0 pφ 1˝φ pxqq, f 0 pφ 1˝ν φ pxqqq ď e r dpφ 1˝φ pxq, φ 1˝ν φ pxqq. Combining this with the previous inequality, we see that the quantity on the right, and hence the left, is uniformly bounded above, independent of x. Therefore ν φ 1˝φ and ν φ 1˝ν φ are bounded distance apart and A 0 is a homomorphism. 5.3. From quasi-isometries to affine homeomorphisms. The flat metric q on E 0 determines an associated affine group AffpE 0 q, and we observe that if φ P Γ is an element of the extension group (which is an isometry ofĒ, and so also a quasi-isometry), we have ν φ P AffpE 0 q. The next step in the proof of rigidity is the following.
Proposition 5.8. For any quasi-isometry φ :Ē ÑĒ, the quasi-isometry ν φ is uniformly close to a unique element ν a φ P AffpE 0 q. The proof of the proposition will take place over the remainder of this subsection. Before getting to the proof, however, we note a useful corollary. Two quasiisometries φ 1 , φ 2 that are a bounded distance apart have ν φ1 and ν φ2 a bounded distance apart, and so by the uniqueness ν a φ1 " ν a φ2 . Thus we have the following.
Fix a triangulation t of X 0 so that the vertex set is the set of cone points and all triangles are Euclidean triangles (that is, they are images of triangles by maps that are locally isometric and injective on the interior; see e.g. [DDLS21, Lemma 3.4]). Moreover, we assume that all saddle connections in some direction α 0 appear as edges of the triangulation. Lift t to a triangulation r t of E 0 . By assumption, all saddle connections in E 0 in direction α 0 are edges of r t, and the complement of the union of this subset is a union of all (interiors of) strips in direction α 0 .
Lemma 5.10. Given a quasi-isometry φ, there is a biLipschitz homeomorphism ν a φ : E 0 Ñ E 0 a bounded distance from ν φ so that ν a φ restricts to an affine map on each triangle of r t. Moreover, if an edge δ of r t has direction α P P, then ν a φ pδq has direction φ P pαq.
We will later prove that ν a φ is in fact globally affine, justifying the notation.
Proof. Given ν " ν φ : E 0 Ñ E 0 , let Bν : S 1 8 Ñ S 1 8 be the restriction of the extension to the Gromov boundary S 1 8 of E 0 . The space G of (unordered) pairs of distinct points in S 1 8 is precisely the space of endpoint-pairs at infinity of unoriented biinfinite geodesics (up to the equivalence relation of having finite Hausdorff distance). The map Bν induces a map Bν˚: G Ñ G.
Let G˚Ă G be the closure of the set of endpoint-pairs at infinity of non-singular geodesics (i.e. geodesics that miss every cone point). Observe that all geodesics in a given strip have the same pair of endpoints, and any geodesic with that pair of endpoints is contained in the strip. Given a strip, we are therefore justified in referring to the pair of endpoints of the strip.
It follows from the description of geodesics with endpoints in G˚(see [BL18, Proposition 2.4]) together with the Veech Dichotomy (see e.g. [MT02]), that for any tξ, ζu P G˚, either tξ, ζu are the endpoints of a strip, or endpoints of a geodesic meeting at most one cone point.
According to Proposition 5.4, for any strip A P S, the strip φ S pAq has finite Hausdorff distance to φpAq, and hence it also has finite Hausdorff distance to νpAq. Since φ S is a bijection, this means that the homeomorphism Bν˚sends the dense subset of G˚consisting of endpoint of strips onto itself, hence ν˚pG˚q " G˚. From this and [BL18, Proposition 4.1] (see also [DELS18,Proposition 11]), it follows that there is a bijection φ Σ0 : Σ 0 Ñ Σ 0 from the set of cone points Σ 0 of E 0 to itself with the following property. If γ Ă E 0 is a geodesic or strip containing x P Σ 0 with endpoints tξ, ζu P G˚, then Bν˚ptξ, ζuq are the endpoints of a geodesic containing φ Σ0 pxq. Given x P Σ 0 consider any two geodesics γ 1 and γ 2 with endpoints in G˚(not necessarily contained in strips) passing through x making an angle at least π{2 with each other. We note that νpxq is contained in νpγ 1 q and νpγ 2 q, and is thus some uniform distance r ą 0 to both of their geodesic representatives. Since γ 1 and γ 2 meet at angle at least π{2, the r-neighborhoods of the geodesic representatives of νpγ 1 q and νpγ 2 q intersect in a uniformly bounded diameter set, which contains φ Σ0 pxq. Therefore, φ Σ0 pxq is uniformly close to νpxq, for all x P Σ 0 .
From the properties of φ Σ0 described above, we see that if x P Σ 0 is contained in a strip A, then φ Σ0 pxq is contained in the strip φ S pAq. For any saddle connection δ in some direction α P P between a pair of points x, y P Σ 0 , there is a unique pair of strips A 1 , A 2 , also in direction α, that contain δ. Since φ Σ0 pxq, φ Σ0 pyq are contained in φ S pA 1 q and φ S pA 2 q, it follows that there is a unique saddle connection with endpoints φ Σ0 pxq, φ Σ0 pyq. For any strip A the saddle connections whose union makes up one of its boundary components is determined by a collection of strips meeting A in the given saddle connections. Considering the cyclic ordering of the endpoints of these strips (and those of A) on S 1 8 , and the fact that Bν is a homeomorphism, it follows that φ Σ0 maps the ordered set of cone points along each boundary component of the strip A by an order preserving (or reversing) bijection to the ordered set of cone points along the boundary components of φ S pAq.
We can now extend the map φ Σ0 to a map ν a φ : E 0 Ñ E 0 using r t as follows. First, recall that any edge of r t is a saddle connection δ connecting two points x, y P Σ 0 . By the previous paragraph, there is a saddle connection δ 1 connecting φ Σ0 pxq and φ Σ0 pyq, and we define ν a φ on δ so that it maps δ by an affine map to δ 1 extending φ Σ0 on the endpoints. This defines ν a φ on the 1-skeleton, r t 1 , and since φ Σ0 is a bounded distance from ν| Σ0 , it follows that ν a φ | r t 1 is a bounded distance from ν| r t 1 . By our assumptions on r t, there is a subset of the edges of r t whose union is precisely the union of boundaries of all strips in direction α 0 . The order preserving (or reversing) property described above for the cone points along the boundary of a strip, together with Proposition 5.4, implies that for any boundary component of any strip A in direction α 0 , ν a φ restricted to its boundary components is a homeomorphism onto the boundary components of φ S pAq. Furthermore, since the sides of any triangle of r t are contained in such a strip A, the ν a φ -image of the sides are contained in φ S pAq. We can now extend ν a φ over the triangles by the unique affine map extending the map on their sides.
Since disjoint strips map to disjoint strips, the map ν a φ is a homeomorphism. By construction, any edge in direction α is sent to an edge in direction φ P pαq. Since r t projects to t, there are only finitely many directions that the sides of a triangle can lie in and so finitely many isometry types of triangles. Each of these finitely many isometry types maps by an affine map to only finitely many types of triangles in the image (because the direction of the images of sides are determined by φ P ), and therefore these affine maps are uniformly biLipschitz. Therefore, ν a φ is biLipschitz, completing the proof.
To show that ν a φ is affine, we analyze the effect of using it to conjugate the action of π 1 S on E 0 .
Lemma 5.11. The action of π 1 S on E 0 obtained by conjugating the isometric action by ν a φ is again an isometric action. Before proving the lemma, we use it to prove the proposition.
Proof of Proposition 5.8. By Lemma 5.11, Λ " ν a φ π 1 Spν a φ q´1 acts by isometries, and ν a φ descends to a homeomorphism µ a φ : S Ñ E 0 {Λ and is biLipschitz with respect to descent to S and E 0 {Λ of q. Since ν a φ and ν are a bounded distance, they have the same boundary maps. Since Bpν a φ q˚" Bν˚maps G˚to G˚, the Current Support Theorem of [DELS18] (and its proof) implies that the descent of µ a φ : pS, qq Ñ pE 0 {Λ, qq is affine. Therefore ν a φ is an affine map which is a bounded distance from ν " ν φ , as required.
Uniqueness follows from the fact that no two distinct affine maps are a bounded distance apart.
Proof of Lemma 5.11. We need to show that for all g P π 1 S, the map ν a φ˝g˝p ν a φ q´1 : E 0 Ñ E 0 is an isometry. For this, fix a triangle τ of r t and consider the restriction to ν a φ pτ q. Let α 1 , α 2 , α 3 P P be the directions of the sides. Setting α 1 i " φ P pα i q, for i " 1, 2, 3, Lemma 5.10 implies that the directions of the sides of ν a φ pτ q are α 1 1 , α 1 2 , α 1 3 . The action of π 1 S on E 0 is by isometries, but it also preserves parallelism (i.e. each element induces the identity on P 1 pqq). Therefore, for any g P π 1 S, the directions of the sides of gpτ q are also α 1 , α 2 , α 3 , and by Lemma 5.10 again, it follows that the sides of ν a φ pgpτ qq are α 1 1 , α 1 2 , α 1 3 . For any g P π 1 S, since ν a φ is affine on τ , the composition ν a φ˝g˝p ν a φ q´1 is also affine on ν a φ pτ q. On the other hand, it also preserves the directions of the sides, α 1 1 , α 1 2 , α 1 3 . Therefore, the restriction of ν a φ˝g˝p ν a φ q´1 is a Euclidean similarity. Triangles of ν a φ p r tq that share a side are scaled by the same factor by the similarity ν a φ˝g˝p ν a φ q´1 in each triangle (since this is the scaling factor on the shared side). Therefore, the similarities agree along edges, and hence ν a φ˝g˝p h a φ q´1 defines a global similarity of E 0 .
So, the action of π 1 S on E 0 obtained by conjugating by ν a φ is an action by similarities. To see that the action is by isometries, suppose that for some element g P π 1 S, the similarity g 0 " ν a φ˝g˝p ν a φ q´1 scales the metric some number λ ‰ 1. Taking the inverse if necessary, we can assume λ ă 1. Fix any x P E 0 and observe that d q pg 0 pxq, g 2 0 pxqq " λd q px, g 0 pxqq, where d q is the distance function on E 0 determined by q. Iterating this, it follows that Since the right-hand side is a convergent geometric series, it follows that tg n 0 pxqu 8 n"1 is a Cauchy sequence. On the other hand, this sequence exits every compact set (since g 0 is an infinite order element of π 1 S), and since q is a complete metric on E 0 , thus we obtain a contradiction. Therefore, the conjugation action of π 1 S is by isometries.
5.4. Injectivity of A. Our next goal is to prove that A : QIpĒq Ñ AffpE 0 q, the homomorphism from Corollary 5.9, is injective. In preparation, it will be useful to have the following general fact about quasiisometries of hyperbolic spaces, whose proof we sketch for convenience of the reader: Lemma 5.12. For each K, C, δ there exists R so that the following holds. Suppose that Z is δ-hyperbolic and that each z P Z lies within δ of all three sides of a nondegenerate ideal geodesic triangle. Let f : Z Ñ Z be a pK, Cq-quasi-isometry that lies within finite distance of the identity. Then f lies within distance R of the identity.
Proof. Since f is within bounded distance of the identity, its extension Bf : BZ Ñ BZ is the identity. Hence if z P Z and ∆ is an ideal geodesic triangle as in the statement, then f p∆q is a pK, Cq-quasigeodesic ideal triangle with the same endpoints as ∆. By the Morse lemma, there is a constant κ " κpK, C, δq ą 0 such that f pzq lies within κ of the three quasi-geodesic sides of f p∆q, and these sides in turn lie within κ of the sides of ∆. Thus z and f pzq both lie within 2κ`δ of all three sides of ∆. Since the set of points within 2κ`δ of all three sides of a nondegenerate geodesic triangle in a δ-hyperbolic space has diameter bounded in terms of κ and δ, we see that d Z pz, f pzqq is bounded solely in terms of δ, K, C, as required.
Proof. We first claim that for any X PD, φpE X q lies within the C 2 -neighborhood of E X , where C 2 " C 2 pφq ą 0. To see this, observe that since ν a φ is the identity and is bounded distance from ν φ , it follows that φ| E0 is within bounded distance of the inclusion of E 0 inĒ. Proposition 5.4 then implies that d Haus pA, φ S pAqq ă`8 for each strip A P S. Since strips that lie within finite Hausdorff distance coincide, we have φ S pAq " A. Combining this fact with Proposition 5.4 it follows that φ P pαpAqq " αpφ S pAqq " αpAq. Hence for each α, we have that φpBB α q lies within Hausdorff distance C of BB α " BB φ P pαq , for C as in Proposition 5.4. Now let X PD be any point and choose distinct α, α 1 P P so that X is contained in the coarse intersection of BB α and BB α 1 , implying that E X lies in the coarse intersection of BB α and BB α 1 . By the coarse preservation of the BB α in the previous paragraph, the coarse intersection of φpBB α q and φpBB α 1 q is within Hausdorff distance C of the coarse intersection of BB α and BB α 1 , and hence E X and φpE X q are within uniform Hausdorff distance. This proves the claim.
Since f X :Ē Ñ E X is e C 2 -bi-Lipschitz when restricted to fibers in the C 2neighborhood of E X , the claim implies that ν X φ " f X˝φ : E X Ñ E X is a quasiisometry with constants depending only on φ and not X. Moreover, since each E X lies within finite (but not necessarily bounded) Hausdorff distance of E 0 , the fact that φ| E0 lies within finite distance of the inclusion E 0 ãÑĒ implies that ν X φ lies within finite distance of the identity E X Ñ E X . Since each E X is uniformly quasiisometric to H 2 , it follows that ν X φ : E X Ñ E X satisfies the assumptions of Lemma 5.12. We conclude that ν X φ is within uniformly bounded distance of the identity for each X PD. Since dpν X φ pxq, φpxqq ď C 2 , it follows that dpx, φpxqq is uniformly bounded, independent of x. This proves the first statement of the proposition.
If Apφq is the identity for some φ P QIpĒq, then by the first part of the proposition, φ is a bounded distance from the identity. Therefore, φ and the identity represent the same class, and A is injective. This completes the proof. 5.5. From affine homeomorphisms to isometries. Next we will choose a particular allowable truncation and construct a homomorphism AffpE 0 q Ñ Isom fib pĒq, that we will eventually show is an isomorphism. We first construct such a homomorphism to the fiber-preserving isometry group of the space E, which avoids the issue of choosing the truncation.
Proof. Recall from §2.1 that the projective tangent space at any non-cone point of E 0 is denoted P 1 pqq and is canonically identified with BD. The derivative of ν : E 0 Ñ E 0 (which may reverse orientations) is a well-defined projective transformation dν P PGLpP 1 pqqq which, using the preferred coordinates on q " q 0 with distinguished vertical and horizontal directions, we canonically identify with PGL 2 pRq. The Teichmüller disk D is the orbit of q under the SL 2 pRq action and is identified with H 2 " SL 2 pRq{SOp2q (see e.g. [DDLS21, §2.8]). As PGL 2 pRq acts isometrically on H 2 , we thus obtain an isometry Φ " dν : D Ñ D whose induced map BΦ of the circle at infinity BD agrees with the derivative dν under the canonical identification BD -P 1 pqq. In particular, setting X " ΦpX 0 q, the geodesic ray in D emanating from X 0 and asymptotic to ξ P P 1 pqq is sent to the geodesic ray emanating from X asymptotic to dνpξq.
We claim the map φ 0 " f X,X0˝ν : E 0 Ñ E X is an isometry of fibers. Indeed, any pair ξ, ξ K P P 1 pqq of orthogonal directions on E 0 are the endpoints of a geodesic ρ in D containing X 0 . Since Φ is an isometry with BΦ " dν, we have that X " ΦpX 0 q lies on the geodesic Φpρq from dνpξ K q to dνpξq; that is, dνpξq, dνpξ K q are orthogonal on X. But since P 1 pqq and P 1 pq X q are canonically identified by the Teichmüller map f X,X0 (see e.g. [DDLS21, §2.8]), this means φ 0 is an affine map whose derivative dφ 0 " dν preserves orthogonality of lines; hence φ 0 is an isometry as claimed. Now we define φ " φ ν : E Ñ E by the formula: φpxq " f Φpπpxqq,X0˝ν˝f0 pxq.
In words, this maps the fiber over a point Y to the fiber over ΦpY q, and the horizontal disk D x , for x P E 0 , to D νpxq . The restriction φ| Dx : D x Ñ D νpxq is an isometry since it covers Φ. To prove that φ is an isometry, it therefore suffices to show that φ| E Y : E Y Ñ E ΦpY q is an isometry for any Y P D.
Fix any Y P D. For Y " X 0 , we have already seen that φ| E0 is the isometry φ 0 : E 0 Ñ E X . If Y ‰ X, there exist unique orthogonal directions α, α K P Ppqq and t ą 0, so that X 0 and Y both lie on the the geodesic from α K to α in D and Y lies distance t from X 0 in the direction of α. This means that f Y,X0 : E 0 Ñ E Y contracts in direction α by e´t and stretches in direction α K by e t . The image ΦpY q lies along the geodesic from dφ 0 pα K q to dφ 0 pαq at distance t from ΦpX 0 q " X; therefore f ΦpY q,X : E X Ñ E ΦpY q contracts by e´t in direction dφ 0 pαq and stretches by e t in direction dφ 0 pα K q. The restriction φ| Y : Y Ñ ΦpY q is given by f ΦpY q,X˝φ0˝fE0,Y . Since φ 0 sends α Þ Ñ dφ 0 pαq and α K Þ Ñ dφ 0 pα K q, the description above shows that φ| Y is an isometry. Therefore φ is an isometry, as required.
To see that ν Þ Ñ φ ν is a homomorphism, note that by construction Φ ν is the unique isometry of D for which BΦ ν " dν. Thus the chain rule implies Φ ν˝g " Φ ν˝Φg is the unique isometry whose action on BD agrees with dpν˝gq " dν˝dg.
Proof. By [DDLS21, Proposition 5.5], Isom fib pEq acts properly discontinuously on E. Therefore E{ Isom fib pEq is a topological orbifold with well-defined, positive Riemannian volume. The index of Γ in Isom fib pEq is the degree of the orbifold cover E{Γ Ñ E{ Isom fib pEq and equals the ratio of the respective volumes. As E{Γ has finite volume, since the quotient D{G has finite area and the fibers E X {π 1 S all have equal, finite area, we conclude that Γ indeed has finite index.
Lemma 5.16. There is an allowable truncationĒ that is Isom fib pEq-invariant and for which restricting toĒ induces an injection Isom fib pEq Ñ Isom fib pĒq.
Remark 5.17. Every fiber-preserving isometry ofĒ uniquely extends to one of E (e.g. by following the proof of Lemma 5.14) and thus Isom fib pEq Ñ Isom fib pĒq is in fact an isomorphism.
Proof. There is a natural map Isom fib pEq Ñ IsompDq that sends Γ onto G. Hence, by the previous lemma, the image G˚of Isom fib pEq under this map contains G with finite index. Therefore G˚acts properly discontinuously on D and we may choose a collection tB α u αPP of 1-separated horoballs as in §2.1 that is G˚-invariant. IfĒ denotes the corresponding truncation of E, it follows that every element of Isom fib pEq preservesĒ. The map Isom fib pEq Ñ Isom fib pĒq given by restricting φ Þ Ñ φ|Ē is injective by [DDLS21, Corollary 5.6] since if φ|Ē is the identity, then φ must be the identity on each Teichmüller disk D x and fiber E X ĂĒ.
Proof. The construction of A in Corollary 5.9 sends the (quasi-)isometry Ψpνq " φ ν |Ē :Ē ÑĒ to the the unique affine homeomorphism of E 0 that is uniformly close to the map f 0˝φν | E0 : E 0 Ñ E 0 . But by the construction of φ ν in Lemma 5.14, f 0˝φν | E0 is affine itself and equal to ν. Thus evidently ApΨpνqq " ν as claimed.
Proof. By construction ν " ν a φ " Apφq is the unique affine homeomorphism bounded distance from f 0˝φ | E0 . As this map is itself affine, we have ν " f 0˝φ | E0 . The isometry Φ : D Ñ D in the construction of Ψpνq is then just the descent of φ to D. Further, for any X, Y P D we have φ| X˝fX,Y " f ΦpXq,ΦpY q˝φ | E Y , since if X lies at distance t ą 0 from Y along the geodesic from α K to α, then both maps send pα K , αq Þ Ñ pBΦpα K q, BΦpαqq while contracting the first by e´t and expanding the second by e t , hence they are the same affine map E Y Ñ E ΦpXq . It follows that the restriction Ψpνq| E Y : E Y Ñ E ΦpY q is then the composition Since this holds for each Y , we conclude Ψpνq " φ as claimed.
It follows that Isom fib pĒq Ñ QIpĒq is injective, since if rφs is the identity in QIpĒq, meaning φ is finite distance from the identity, then Apφq " Aprφsq and consequently φ " ΨpApφqq are both the identity. Finally, IsompĒq Ñ QIpĒq is injective since we have Isom fib pĒq " IsompĒq by [DDLS21, Corollary 5.4]. 5.6. Rigidity. We are now ready to complete the proof of Theorem 1.7: Proof of Theorem 1.7. By Lemma 5.18, the composition AffpE 0 q Ψ Ñ Isom fib pĒq Ñ IsompĒq Ñ QIpĒq -QIpΓq A Ñ AffpE 0 q is the identity. Hence the first map Ψ is injective, and the remaining maps are injective by Lemma 5.19 and Proposition 5.13 . It follows that each map above is an isomorphism, as claimed. The fact that Γ has finite index in IsompĒq -QIpΓq thus follows from Lemma 5.15.
Standard techniques (see, for example [Sch95,§10.4]) now imply the following: Corollary 5.20. If H is any finitely generated group quasi-isometric to Γ, then H and Γ are weakly commensurable, meaning H has a finite normal subgroup N so that H{N and Γ contain finite index subgroups that are isomorphic.
This proof requires one more lemma.
Lemma 5.21. For every K there exists R 1 such that if φ :Ē ÑĒ is a pK, Kqquasi-isometry that lies within finite distance of the identity, then φ lies within distance R 1 of the identity, meaning dpx, φpxqq ď R 1 for all x PĒ.
Proof. First define a mapφ :D ÑD by settingφpXq " Y , where Y is the point provided by Proposition 5.4 such that d Haus pφpE X q, E Y q ď C. Since dDpX, Y q " dĒpE X , E Y q for all X, Y inD, we see thatφ is a quasi-isometry with constants depending only on K. It also lies within finite distance of the identity, as it inherits this property from φ; thus applying Lemma 5.12 to Z "D implies thatφ lies within uniformly finite distance of the identity. That is, there exists R depending only on K so that d Haus pE X , ΦpE X qq ď R for all X PD. Hence, Lemma 5.6 implies that for each map ν X φ " f X˝φ | E X is a pK 1 , K 1 q-quasi-isometry for some K 1 depending only on K. Again by Lemma 5.12, this time with Z " E X , we see that each ν X φ moves points uniformly bounded distance, and therefore φ| E X lies within uniform distance of the inclusion of E X inĒ. Since this holds for all X, we have that φ lies within uniform distance of the identity, as required.
Proof of Corollary 5.20. If H is quasi-isometric to Γ, there is a quasi-isometry µ : H ÑĒ with a quasi-inverse µ´1 :Ē Ñ H. Left multiplication by h P H gives an isometry L h : H Ñ H. In this way, for each h P H we obtain a quasiisometry Bphq " µ˝L h˝µ´1 ofĒ with uniformly bounded constants. Let us also set B 1 phq " ΨpApBphqqq P Isom fib pĒq " IsompĒq, which is the unique isometry ofĒ at finite distance from Bphq. Since the quasi-isometry constants of Bphq are uniform, depending only on µ, it follows from Lemma 5.21 that there is a constant R 1 so that dpBphqpxq, B 1 phqpxqq ď R 1 for all x PĒ and h P H.
We now claim the homomorphism B 1 : H Ñ IsompĒq has finite kernel and cokernel. Indeed, if B 1 phq " IdĒ the above implies Bphq moves µpeq (and in fact all points) distance at most R 1 . But this means L h moves the identity e P H uniformly bounded distance, and there are only finitely many such elements of H. To prove B 1 has finite cokernel it suffices, as in Lemma 5.15, to showĒ{B 1 pHq has finite volume or, better yet, finite diameter. For this, given x, y PĒ we must find h so that dpB 1 phqpxq, yq is uniformly bounded. This is equivalent to bounding dpBphqpxq, yq " dpµph¨µ´1pxqq, yq, which is coarsely d H ph¨µ´1pxq, µ´1pyqq. Since H acts transitively on itself, this is clearly possible.
We now see that H{ kerpB 1 q and Γ are both realized as finite index subgroups of IsompĒq and hence that their intersection has finite index in both.