Character varieties of a transitioning Coxeter 4-orbifold

In 2010, Kerckhoff and Storm discovered a path of hyperbolic 4-polytopes eventually collapsing to an ideal right-angled cuboctahedron. This is expressed by a deformation of the inclusion of a discrete reflection group (a right-angled Coxeter group) in the isometry group of hyperbolic 4-space. More recently, we have shown that the path of polytopes can be extended to Anti-de Sitter geometry so as to have geometric transition on a naturally associated 4-orbifold, via a transitional half-pipe structure. In this paper, we study the hyperbolic, Anti-de Sitter, and half-pipe character varieties of Kerckhoff and Storm's right-angled Coxeter group near each of the found holonomy representations, including a description of the singularity that appears at the collapse. An essential tool is the study of some rigidity properties of right-angled cusp groups in dimension four.

In his thesis [Dan11], Danciger showed that when the limit is 2-dimensional and hyperbolic, it often regenerates to Anti-de Sitter (AdS) structures as well, so as to have geometric transition from hyperbolic to AdS structures (see also [Dan13,Dan14,AP15,FS19,Tre19]). To that purpose, he introduced half-pipe (HP) geometry, which is a limit geometry [CDW18] of both hyperbolic and AdS geometries inside projective geometry, and encodes the behaviour of such a collapse "at the first order". One can indeed suitably "rescale" the structures inside the "ambient" projective geometry along the direction of collapse, so as to get at the limit a 3-dimensional "transitional" HP structure.
Concerning dimension four, Kerckhoff and Storm [KS10] described a path t → P t , t ∈ (0, 1], of hyperbolic 4-polytopes which collapse as t → 0 to a 3-dimensional ideal right-angled cuboctahedron. This induces a path of incomplete hyperbolic structures on a naturally associated 4-orbifold O. The orbifold fundamental group of O is a rank-22 right-angled Coxeter group Γ 22 , which embeds in Isom(H 4 ) as a discrete reflection group when t = 1. In [RS] (see also [Sep19]), we found a similar path of AdS 4-polytopes such that the two Let us include some comments to elucidate the content of Theorem 1.1. First, our proofs actually show that the representation ρ 0 has a neighbourhood in Hom(Γ 22 , G) that is homeomorphic to (H ∪ V) × G + , in such a way that the action of G + corresponds to obvious left multiplication by G + on the second factor (see Remark 4.5).
Let U, V and H be the preimages in Hom(Γ 22 , G) of U, V and H, respectively. By "smoothness" of the "components" V and H of U we actually refer to V, H and U, respectively. In particular, V and H are smooth manifolds (of dimension 11 and 22, respectively). The smoothness of V and H together with the local product structure in a neighbourhood of ρ 0 induce a smooth structure on the components H and V in the quotient.
The "transversality" of V and H is defined as follows: V ∩ H is the G-orbit of ρ 0 , and the Zariski tangent spaces of V and H intersect transversely in the Zariski tangent space of Hom(Γ 22 , G) at ρ 0 (and hence at any other point of its orbit). In particular, every infinitesimal deformation tangent to both V and H is tangent to the G + -orbit of ρ 0 . (See Section 1.5 below and Remark 7.9 for more details.) Our analysis will also show that, when G is Isom(H 4 ) or Isom(AdS 4 ), the character variety X(Γ 22 , G) is homeomorphic to the GIT quotient Hom(Γ 22 , G)/ /G + near each [ρ t ] (see Remark 4.6). In other words, X(Γ 22 , G) is Hausdorff near [ρ t ]. Moreover, the natural smooth structure of each component is coherent with the real semialgebraic structure of the GIT quotient (see also Remark 7.9).
In order to further discuss our results, we first need to describe the three paths of geometric representations of Theorem 1.1.
1.2. The three deformations. For t = 1, the hyperbolic polytope P 1 ⊂ H 4 is obtained in [KS10] from the ideal right-angled 24-cell by removing two opposite bounding hyperplanes. So P 1 has two "Fuchsian ends", and in particular its volume is infinite. The reflection group Γ 22 < Isom(H 4 ) associated to P 1 is thus a right-angled Coxeter group obtained by removing from the reflection group Γ 24 of the ideal right-angled 24-cell two generators (reflections at two opposite facets).
As a sort of "reflective hyperbolic Dehn filling", Kerckhoff and Storm show that the inclusion Γ 22 → Isom(H 4 ) is not locally rigid. This is done by moving the bounding hyperplanes of P 1 in such a way that the orthogonality conditions given by the relations of Γ 22 are maintained, and thus obtaining a path ρ H 4 t of geometric representations of Γ 22 . As t decreases from 1, the combinatorics of P t changes a few times, until the volume of P t becomes finite. Most of the dihedral angles of P t are constantly right, while the varying ones are all equal and tend to π as t → 0, when P t collapses to the cuboctahedron. As an abstract group, Γ 22 can be identified to the orbifold fundamental group of an orbifold O supported on the complement in P t of the ridges with non-constant dihedral angle.
Kerckhoff and Storm show moreover that the space of conjugacy classes of representations Γ 22 → Isom(H 4 ) deforming the inclusion is a smooth curve outside of the collapse. In other words, for t = 0 the only non-trivial deformation (up to conjugacy) is given by the found holonomies ρ H 4 t . In [RS], we produced a path of AdS 4-polytopes with the same combinatorics of the hyperbolic polytope P t for t ∈ (0, ε), such that the same orthogonality conditions between the bounding hyperplanes are satisfied, and again collapsing to an ideal right-angled cuboctahedron in a spacelike hyperplane H 3 of AdS 4 . Some bounding hyperplanes are spacelike, and some others are timelike. We have in particular a path of AdS orbifold structures on O, with holonomy representation ρ AdS 4 t : Γ 22 → Isom(AdS 4 ) given by sending each generator to the corresponding AdS reflection.
We moreover find in [RS] a one-parameter family of transitional HP structures on O, with holonomy ρ HP 4 t . To interpret distinct elements in this family, recall that in half-pipe space there is a preferred direction under which the HP metric is degenerate. An HP structure is never rigid, because one can always conjugate with a transformation which "stretches" the degenerate direction, and obtain a new structure equivalent to the initial one as a real projective structure, but inequivalent as a half-pipe structure. We discover here (this is part of the content of Theorem 1.1) that such stretchings are the only possible deformations, so that the found HP structures are essentially unique.
Finally, we remark that the geometric transition described in [RS] induces a continuous deformation connecting in the PGL(5, R)-character variety "half" of the path in V ⊂ X(Γ 22 , Isom(H 4 )) (which is exactly the path of hyperbolic representations exhibited by Kerckhoff and Storm, for t ∈ (0, 1]) and "half" of the analogous path in X(Γ 22 , Isom(AdS 4 )), going through a single half-pipe representations ρ HP 4 t0 with t 0 = 0 (this value of t 0 can be chosen arbitrarily, up to reparameterising the entire deformation).
1.3. About the result. Theorem 1.1 contains several novelties. First, while the smoothness of the Isom(H 4 )-character variety for t > 0 was proved in [KS10], the smoothness on the AdS and HP sides is completely new. Second, the study of the character variety at the "collapsed" point [ρ 0 ] is a new result in all three settings. Some motivations follow.
First of all, we found worthwhile analysing the behaviour of the deformation space of the AdS orbifold O -equivalently, the Isom(AdS 4 )-character variety of Γ 22 near [ρ t ] -and compare it with the hyperbolic counterpart. In fact, the literature seems to miss a study of deformations of AdS polytopes in this spirit. With respect to hyperbolic geometry, one may expect more flexibility in AdS geometry, but we find that the behaviour on the AdS side is the same as the hyperbolic counterpart. (In regard, see however [BBD + 12, Question 9.3] and the related discussion.) Regarding the collapse, in [KS10,Section 14] Kerckhoff and Storm mention that the family of hyperbolic polytopes P t ⊂ H 4 , t > 0, is expected to have interesting geometric limits by rescaling in the direction transverse to the collapse. On the other hand, they assert that "providing the details of this geometric construction would require more space than perhaps is merited here".
The work [RS] provides a complete description of such a geometric limit in half-pipe geometry, which, after the work of Danciger, seems the best suited in order to analyse this kind of collapse. One can in fact consider the limiting HP structure as an object which encodes the collapse at the first order, essentially keeping track of the derivatives of all the associated geometric quantities. Thus, if [RS] describes the collapse at level of geometric structures, Theorem 1.1 (in the hyperbolic and AdS setting) gives a precise description of the collapse at level of the character variety.
Finally, our study of the HP character variety shows that the only deformations of the found half-pipe orbifold structure are obtained by "stretching" in the degenerate direction. The presence of many commutation relations forces the rigidity of the HP structures.
Together with the hyperbolic and AdS picture, this shows that "nearby" there is no collapsing path of hyperbolic or AdS orbifold structures other than the ones we found (up to reparameterisation). This should be compared with some 3-dimensional examples found by Danciger [Dan13,Section 6], where the transitional HP structure deforms non-trivially to nearby HP structures that regenerate to non-equivalent AdS structures, despite not regenerating to hyperbolic structures.
All in all, Theorems 1.1 exhibits a strong lack of flexibility around this example. Its proof, explained in the next section, suggests that this could be more generally due to dimension issues, confirming the usual feeling that "the rigidity increases with the dimension".
An overview of the ideas behind the proof of Theorem 1.1 follows. We start in Section 1.4 with the geometric tools, which lead to the topological description of the neighbourhood U of [ρ 0 ]. In Section 1.5, we then outline the algebraic aspects, which lead to the description of the Zariski tangent space and the transversality statement.
1.4. Cusp rigidity in dimension four. The holonomy representations ρ t have the property that each generator in the standard presentation of Γ 22 is sent by ρ t to a (hyperbolic or AdS) reflection, and this property is preserved by small deformations. As in [KS10], we thus reduce to studying the configurations of hyperplanes of reflections satisfying certain orthogonality conditions. Once this set-up is established, there are two main facts to prove: the smoothness of the character variety outside the collapse, and the description of the collapse itself.
For the first fact, the proof on the AdS side follows the general lines of the proof given in [KS10] for the hyperbolic case. However, different arguments are required for one point of fundamental importance concerning a property of rigidity of cusp representations in dimension four.
In fact, in [KS10] a preliminary lemma is proved, which can be summarised by saying that in dimension 4 "cusp groups stay cusp groups". More precisely, if we consider the orbifold fundamental group of a Euclidean cube Γ cube , this property states that any representation of Γ cube into Isom(H 4 ) sending the six standard generators to reflections in six distinct hyperplanes with the property of being asymptotic to a common point at infinity (a "cusp group") can only be deformed by preserving this property. Note that the analogue fact is false in dimension three, where the situation is more flexible.
We do prove the analogous property for Anti-de Sitter geometry in dimension 4 (Proposition 3.15), where a cusp group is defined analogously. There are however remarkable differences due to the different nature of hyperbolic and AdS geometries, for instance a cusp group in AdS 4 will be generated by 4 reflections in timelike hyperplanes and 2 reflections in spacelike hyperplanes. The proof of this rigidity property in AdS uses therefore ad hoc arguments and is somehow more surprising than its hyperbolic counterpart, as in general a little more flexibility might be expected for AdS geometry.
Once this fundamental property is established, the proof of the smoothness of the curve is based on a careful analysis of the structure of the group Γ 22 and the possible deformations of the polytope P t , relying on the application of the above rigidity property to each peripheral subgroup (there is a cusp group Γ cube < G associated to each ideal vertex of P t ). The methods are rather elementary, although some intricate computation is necessary, and the general strategy is similar to that of the hyperbolic analogue provided in [KS10].
Let us now explain our arguments to analyse the collapse in both the H 4 and AdS 4 character variety. The proof is essentially the same for both cases, so let us focus on the hyperbolic case (that is, G = Isom(H 4 )) in this introduction for definiteness.
It is not difficult to describe the two components V and H of the neighbourhood U from a geometric point of view: the "vertical" curve V consists of the conjugacy classes of the holonomy representations ρ G t of O, t > 0, plus the natural extension of the path for t < 0 given by r • ρ G −t • r. Here r is the reflection in the totally geodesic copy of H 3 to which the polytope collapses as t = 0. On the other hand, the "horizontal" 12-dimensional component H consists of representations which fix setwise this copy of H 3 , and deform the reflection group of the ideal right-angled cuboctahedron in H 3 .
One then has to show that there exists a neighborhood of [ρ 0 ] such that every point in this neighborhood belongs to one of these two components -namely, there are no other conjugacy classes of representations nearby [ρ 0 ]. To prove this, we refine the study of the rigidity properties of the cusps. We introduce a notion of collapsed cusp group: a representation of Γ cube defined similarly to cusp groups, but allowing that two generators are sent to reflections in the same hyperplane. The restriction of ρ 0 to each peripheral subgroup is in fact a collapsed cusp group. Then we prove a more general version of the aforementioned rigidity property "cusp groups stay cusp groups", by showing that "collapsed cusp groups either stay collapsed, or deform to cusp groups". More precisely, representations nearby a collapsed cusp group either keep the property that two opposite generators are sent to the same reflection, or they become cusp groups in the usual sense.
By an analysis of the character variety in the spirit of Kerckhoff and Storm, we show that the "vertical" curve V is smooth also at t = 0 if we impose that the asymptoticity conditions are preserved. Applying the more general property of rigidity which includes the "collapsed" case is then the fundamental step to conclude the proof.
Concerning the proof for the half-pipe case, it follows a similar line, but many steps are dramatically simpler. The key point is again a rigidity property for four-dimensional (collapsed) cusp groups, which is showed rather easily by using the isomorphism between G HP 4 and the group of isometries of Minkowski space R 1,3 , which is a semidirect product O(1, 3) R 1,3 . The proof then parallels the steps for the hyperbolic and AdS case, except that the smoothness of the vertical component V is granted by the fact that -thanks to this semidirect product structure of G HP 4 -V identifies with the first cohomology vector space H 1 ρ0 (Γ 22 , R 1,3 ). The proof that this vector space is 1-dimensional (see (2) below) requires a certain amount of technicality and relies on a precise study of the group-theoretical structure of Γ 22 .
1.5. The Zariski tangent space and the first cohomology group. Let g be the Lie algebra of G, and Ad : G → Aut(g) be the adjoint representation. The Zariski tangent space to Hom(Γ 22 , G) at ρ is naturally identified to the space Z 1 Ad ρ (Γ 22 , g) of cocycles with coefficients twisted by Ad • ρ. Roughly speaking, the cohomology group H 1 Ad ρ (Γ 22 , g) plays the same role for the quotient X(Γ, G) at [ρ]. Indeed the coboundaries B 1 Ad ρ (Γ 22 , g) are precisely the infinitesimal deformations tangent to the orbit. See [Wei64,JM87] for more details.
Since ρ 0 preserves a totally geodesic copy of H 3 , we have a natural decomposition: We show that the vector space H 1 Ad ρ0 (Γ 22 , g) is 13-dimensional. In the decomposition (1), the first factor is 12-dimensional and "tangent" to H, while the second factor is 1dimensional and "tangent" to V; moreover integrable vectors are precisely those lying in one of these two factors. This statement is made more precise by looking at the representation variety Hom(Γ 22 , G), where the tangent spaces of the smooth varieties V and H are generated by the preimages in Z 1 Ad ρ0 (Γ 22 , g) of the two factors in the decomposition (1). Hence the intersection V ∩ H is transverse and consists precisely of the orbit of ρ 0 .
As mentioned above, we prove (in Proposition 6.5) that the second factor in the decomposition (1) has dimension 1, namely The non-trivial elements in this vector space are obtained geometrically from the half-pipe holonomy representations that we constructed in [RS], and are easily shown to be "first-order deformations" of paths in V. On the other hand, the first factor in the decomposition (1) is 12-dimensional by general reasons, namely by an orbifold version of hyperbolic Dehn filling (note that the cuboctahedron has exactly 12 vertices). Its elements are again integrable and tangent to deformations in H. This cohomological computation is the main algebraic step in the proof of Theorem 1.1. As another noteworthy comment on the consequences of (2), recall Danciger's result [Dan13, Theorem 1.2]: the existence of geometric transition on a compact half-pipe 3manifold X , with singular locus a knot Σ, is proved under the sole condition Now, for any representation ρ : Γ → O(1, 2) with Γ finitely generated there is a natural identification H 1 Ad ρ (Γ, so(1, 2)) ∼ = H 1 ρ (Γ, R 1,2 ) . In presence of a collapse of hyperbolic or AdS structures of dimension n, the holonomy representations of a rescaled HP structure naturally provide elements of the cohomology group H 1 ρ0 (π 1 (X Σ), R 1,n−1 ). In particular, the correct generalisation of Danciger's condition (3) to any dimension n would be: in agreement with what we found for Γ 22 (the orbifold fundamental group of O) -compare with (2). In conclusion, this suggests that a higher-dimensional regeneration result in the spirit of Danciger, although far from reach at the present time, might be reasonable.
Organisation of the paper. In Section 2, we establish the set-up for the proof of Theorem 1.1 in the hyperbolic and AdS cases. In Section 3, which may be of independent interest, we study deformations of some right-angled Coxeter groups represented as "cusp groups" in Isom(H n ) and Isom(AdS n ) for n = 3, 4. In Section 4, we study the Isom(H n ) and Isom(AdS n ) character varieties of Γ 22 , concluding the first part of the proof of Theorem 1.1 for the hyperbolic and AdS case. Section 5 is the analogue of Sections 2 and 3 for half-pipe geometry. In Section 6 we provide the explicit computation of (2), and use it for the first part of the proof of Theorem 1.1 in the HP case. In Section 7 we conclude the proof of Theorem 1.1 in all the three cases, relying on algebraic computations of the first cohomology group.

Reflections in hyperbolic and AdS geometry
In this section we establish the set-up for the study of hyperbolic and AdS character varieties of right-angled Coxeter groups.
2.1. Hyperbolic and AdS geometry. We begin with the necessary definitions and notation. Let q ±1 be the quadratic form on R n+1 of signature (−, +, . . . , +, ±) defined by: n−1 ± x 2 n , and let b ±1 be the associated bilinear form.
The n-dimensional hyperbolic space H n is defined via the "hyperboloid model" as: The restriction of q 1 endows H n with a complete Riemannian metric of constant negative sectional curvature, whose isometry group Isom(H n ) is identified to an index-two subgroup of O(1, n), namely the subgroup of those linear isometries of q 1 which preserve H n . Despite this subgroup is defined by an inequality, if n is even Isom(H n ) is naturally isomorphic to the algebraic Lie group SO(1, n) (via A → A if A preserves H n ⊂ R n+1 , and A → −A otherwise).
The boundary at infinity of H n is the projectivisation of the cone of null directions for the quadratic form q 1 : Hence both H n and ∂H n can be seen as subsets of RP n . The topology of RP n induces a natural topology on H n ∪ ∂H n , which makes it homeomorphic to the closed n-ball D n . Finally, given a subset A ⊂ H n that is closed as a subspace of H n , its ideal closure A is the closure of A in RP n . In particular, we have H n = H n ∪ ∂H n . The n-dimensional Anti-de Sitter space is defined as: and the restriction of q −1 endows AdS n with a Lorentzian metric of constant negative sectional curvature. Observe that AdS n is homeomorphic to S 1 × R n−1 . Its isometry group Isom(AdS n ) is identified to the algebraic Lie groups O(q −1 ) ∼ = O(2, n − 1). The boundary at infinity of AdS n is defined as the image of the null directions for the quadratic form q −1 , now in the projective sphere RP n := (R n+1 {0})/R >0 : Interpreting AdS n and ∂AdS n as subsets of RP n gives AdS n ∪ ∂AdS n a natural topology, which makes it homeomorphic to S 1 × D n−1 . Again, the ideal closure of a subset A ⊂ AdS n that is closed in AdS n is its closure A in RP n , and we have AdS n = AdS n ∪ ∂AdS n .

Hyperplanes and reflections.
A hyperplane of H n (resp. AdS n ) is the intersection, when non-empty, of a linear hyperplane of R n+1 with H n (resp. AdS n ). Notice that, unlike the hyperbolic case, the intersection of a linear hyperplane with AdS n is always non-empty, and sometimes disconnected (see the discussion preceding Lemma 2.2). In both cases hyperplanes are totally geodesic. A reflection in hyperbolic, resp. Anti-de Sitter, geometry is a non-trivial involution r ∈ Isom(H n ), resp. Isom(AdS n ), that fixes point-wise a hyperplane.
Let us denote by ⊥ 1 the orthogonal complement with respect to the bilinear form b 1 , and let X ∈ R n+1 be a vector. The linear hyperplane X ⊥1 of R n+1 intersects H n if and only if q 1 (X) > 0, i.e. if X is spacelike for q 1 . Hence to every q 1 -spacelike vector X is associated a hyperplane It is clearly harmless to assume that q 1 (X) = 1, so that the vector X is uniquely determined up to changing the sign. Given a hyperplane H X in H n , there is a unique reflection r X fixing the given hyperplane. Indeed, the reflection r X is the linear transformation in O(q 1 ) which fixes X ⊥1 and acts on the subspace generated by X as minus the identity. Two spacelike unit vectors X and Y give the same reflection if and only if X = ±Y . Finally, it is a simple exercise to show that two reflections r X and r Y commute if and only if either X = ±Y or X and Y are orthogonal for the bilinear form b 1 .
We summarise the above considerations in the following statement: Lemma 2.1. There is a two-sheeted covering map which maps a spacelike unit vector X to the unique reflection r X fixing H X point-wise. Moroever, two distinct reflections r X and r Y commute if and only if b 1 (X, Y ) = 0.
The subset of vectors in R n+1 such that q 1 (X) = +1 is usually called de Sitter space.
Let us now move to Anti-de Sitter geometry. A hyperplane is called spacelike, timelike or lightlike if the induced bilinear form, obtained as the restriction of the Lorentzian metric of AdS n , is positive definite, indefinite or degenerate, respectively. Spacelike hyperplanes are disconnected, and each of the two connected components is an isometrically embedded copy of H n−1 . Timelike hyperplanes are isometrically embedded copies of AdS n−1 .
Let us denote by ⊥ −1 the orthogonality relation with respect to b −1 . We have: Lemma 2.2. Given a vector X ∈ R n+1 , the intersection X ⊥−1 ∩ AdS n is non-empty, and is: The hyperplane of fixed points of an AdS reflection is either spacelike or timelike. Similarly to the hyperbolic case, given a vector X such that q −1 (X) = ±1, the unique reflection fixing H X = X ⊥−1 ∩ AdS n is induced by the linear transformation in O(q −1 ) acting on X ⊥−1 as the identity and on Span(X) (which is in direct sum with X ⊥−1 since q −1 (X) = 0) as minus the identity. like hyperplane, at a point at infinity, or be disjoint. The picture (n = 3) is in an affine chart for the real projective sphere, where Anti-de Sitter space is the interior of a onesheeted hyperboloid. Each disk drawn in the picture represents a connected component of a spacelike hyperplane, and has an isometric copy on the "opposite" affine chart of RP n , obtained by applying minus the identity to AdS n .
In conclusion, we have another summarising statement: There is a two-sheeted covering map which maps a spacelike or timelike unit vector X to the unique reflection r X fixing H X pointwise. Moroever, two distinct reflections r X and r Y commute if and only if b −1 (X, Y ) = 0.
The space {X ∈ R n+1 : q −1 (X) = ±1} has two connected components, as well as the space of reflections. The component defined by q −1 (X) = −1 is a copy of Anti-de Sitter space itself.
2.3. Relative position of hyperplanes. It will be useful to discuss the relative position of hyperplanes. For hyperbolic geometry, this is easily summarised: Lemma 2.4. Let H X and H Y be two distinct hyperplanes in H n , for q 1 (X) = q 1 (Y ) = 1. Then the following hold: In the second item of Lemma 2.4, H X and H Y intersect in exactly one point at infinity p ∈ ∂H n . In this case, we say that H X and H Y are asymptotic (at p). By little abuse, sometimes we also say that a hyperplane H is asymptotic to a point at infinity p if p ∈ H. Note the difference between the two notions: if the first item holds, then H X and H Y are not asymptotic, despite being asymptotic to p for any point at infinity p ∈ H X ∩ H Y = ∅.
For Anti-de Sitter hyperplanes, it is necessary to distinguish several cases. Here we will only consider the cases of interest for the proofs of our main results. For spacelike hyperplanes, we have (see Figure 2): Similar terminology for asymptoticity is adopted when the second item of Lemma 2.5 occurs, the only difference with the hyperbolic case being that now H X and H Y intersect in exactly two antipodal points at infinity ±p ∈ ∂AdS n .
For two AdS timelike hyperplanes the situation is different, as explained in the following lemma. See also Figure 3.
Lemma 2.6. Let H X and H Y be two distinct timelike hyperplanes in AdS n , for q −1 (X) = q −1 (Y ) = 1. Then H X and H Y intersect in AdS n and the intersection H X ∩ H Y is: • spacelike if and only if |b −1 (X, Y )| > 1; • lightlike if and only if |b −1 (X, Y )| = 1; • timelike if and only if |b −1 (X, Y )| < 1.
In the first case of Lemma 2.5 and Lemma 2.6, the intersection H X ∩ H Y consists of two totally geodesic copies of H n−2 ; in the third case of Lemma 2.6 it is a totally geodesic copy of AdS n−2 .

Right-angled cusp groups in hyperbolic and AdS geometry
In this section, which may be of independent interest, we study deformations of some right-angled Coxeter groups represented as "cusp groups" in Isom(H n ) and Isom(AdS n ), for n = 3, 4. Let us begin in the next subsection with a general set up.
3.1. Coxeter groups and representation varieties. Given a finitely presented group Γ and an algebraic Lie group G, we denote by Hom(Γ, G) the space of representations ρ : Γ → G. Since Hom(Γ, G) is naturally an affine algebraic set (see also Section 7.1), it is called representation variety.
In the remainder of the paper, we will restrict the attention to the case where Γ is a right-angled Coxeter group, which we now define.
Definition 3.1 (RACG). Given a finite set S and a subset R of (unordered) pairs of distinct elements of S, the associated right-angled Coxeter group has presentation: For instance, the group generated by the reflections in the sides of a right-angled Euclidean or hyperbolic polytope is a right-angled Coxeter group.
We will only be interested in representations of a right-angled Coxeter group Γ which send every generator to a reflection. Let us introduce more formally this space.
Definition 3.2 (The set Hom refl ). Let G be Isom(H n ) or Isom(AdS n ) and Γ a right-angled Coxeter group as above. We define Hom refl (Γ, G) as the subset of Hom(Γ, G) of representations ρ such that: • for every s ∈ S, the isometry ρ(s) is a reflection, and • for every (s 1 , s 2 ) ∈ R, the reflections ρ(s 1 ) and ρ(s 2 ) are distinct.
Reflections constitute a connected component in the space of order-two isometries in Isom(H n ), while in Isom(AdS n ) they constitute two connected components, given by reflections in spacelike and timelike hyperplanes. Moreover, by Lemmas 2.1 and 2.3, two distinct reflections commute if and only if their fixed hyperplanes are orthogonal. Hence we immediately get: To simplify the computations, we will follow [KS10] and adopt a local model for the representation variety which is well adapted to our setting. Roughly speaking, we only consider the deformations of the hyperplanes fixed by the reflection associated to each generator. This will reduce significantly the complexity of the problem, since (in dimension n) for each generator we have a vector of n + 1 entries (giving the hyperplane of reflection) in place of an (n + 1) × (n + 1) matrix (giving the reflection itself).
More precisely, the following lemma gives a local parametrisation of the set Hom refl (Γ, G): Lemma 3.4. The set Hom refl (Γ, G) is finitely covered by a disjoint union of subsets of R (n+1)|S| defined by the vanishing of |S| + |R| quadratic conditions.
Proof. Let us first give the proof for G = Isom(H n ). Let us identify R (n+1)|S| to the vector space of functions f : S → R n+1 . For every representation ρ : Γ → G in Hom refl (Γ, G), we can choose a function f such that q 1 (f (s)) = 1 and ρ(s) = r f (s) for every generator s. In fact there are 2 |S| possible choices of such an f , differing by changing sign to the image of each generator, and they all satisfy the following conditions: (1) The vector f (s) is unitary, meaning that q 1 (f (s)) = 1, hence giving |S| quadratic conditions. (2) For each of the commutation relations s i s j = s j s i in Γ, by Lemma 2.1 the corresponding vectors f (s i ) and f (s j ) are orthogonal with respect to b 1 .
Conversely, every f satisfying these conditions induces the representation ρ in Hom refl (Γ, G) defined by ρ(s) = r f (s) . Define a function We have shown that g −1 (0) is a 2 |S| -sheeted covering of Hom refl (Γ, Isom(H n )), with deck transformations given by the group (Z/2Z) |S| . The proof for the Anti-de Sitter case is analogous, except that we have to distinguish several cases, depending on whether ρ(s) is a reflection in a spacelike or timelike hyperplane. In the former case, we must impose q −1 (f (s)) = −1, and in the latter q −1 (f (s)) = 1 (see Lemma 2.2). The orthogonality conditions are the same, but now using the bilinear form b −1 , by Lemma 2.3. In conclusion we have that Hom refl (Γ, Isom(AdS n )) is finitely covered by a disjoint union of |S| subsets each defined by the vanishing of a quadratic function g : R (n+1)|S| → R |S|+|R| . 3.2. Flexibility in dimension three. Let Γ rect denote the right-angled Coxeter group generated by the reflections along the sides of a Euclidean rectangle. The standard presentation of Γ rect has 4 generators (one for each side of the rectangle), and relations such that each generator has order two and reflections in adjacent sides commute. We will also consider other similar representations of Γ rect , which occur in correspondence to a collapse, when two non-commuting generators are sent to the same reflection. Let us begin with the hyperbolic case: Definition 3.7 (Collapsed cusp group for H 3 ). The image of a representation of Γ rect into Isom(H 3 ) is called a collapsed cusp group if the four generators are sent to reflections along three distinct planes asymptotic to a common point at infinity.
Let ρ be a representation near a given ρ ∈ Hom refl (Γ rect , Isom(H 3 )), and s be a generator of Γ rect . In virtue of Lemma 3.3 and the discussion below, we refer to the fixed-point set of ρ (s) as a plane of ρ .
In [KS10,Lemma 5.1], the following property of cusp groups is proved: Proposition 3.8. Let ρ : Γ rect → Isom(H 3 ) be a representation whose image is a cusp group. For all nearby representations whose image is not a cusp group, a pair of opposite planes intersect in H 3 , while the other pair of opposite planes have disjoint ideal closures in H 3 .
In fact, a simple adaptation of the proof shows: Proposition 3.9. Let ρ : Γ rect → Isom(H 3 ) be a representation whose image is a cusp group or a collapsed cusp group. For all nearby representations ρ , exactly one of the following possibilities holds: (1) If s 1 and s 2 are generators such that ρ(s 1 ) = ρ(s 2 ), then ρ (s 1 ) = ρ (s 2 ).
(2) The image of ρ is a cusp group.
(3) A pair of opposite planes intersect in H 3 , while the other pair of opposite planes have disjoint ideal closures in H 3 .
The first case may hold only if ρ is a collapsed cusp group. Under this hypothesis, Proposition 3.9 can be rephrased by saying that a deformation of a collapsed cusp group either preserves the property that two planes corresponding to non-adjacent sides of the rectangle coincide (which is the case when the collapsed cusp group remains a collapsed cusp group, for instance), or it falls in the class of representations described in Proposition 3.8, namely, deformations of non-collapsed cusp groups. If ρ is a cusp group, then the content of Proposition 3.9 is the same as Proposition 3.8.
We now move to the AdS version of Propositions 3.8 and 3.9, for which we will give a complete proof. Proofs for the hyperbolic case are easier and can be repeated by mimicking the AdS case.
Note that for an AdS cusp group the four planes necessarily satisfy the orthogonality conditions as in a rectangle, and therefore two of them are spacelike and two timelike. We will show the following proposition, which is the AdS version of Proposition 3.8.
Proposition 3.10. Let ρ : Γ rect → Isom(AdS 3 ) be a representation whose image is a cusp group. For all nearby representations whose image is not a cusp group, exactly one of the following possibilities holds: (1) The ideal closures of the two spacelike planes are disjoint in AdS 3 , whereas the two timelike planes intersect in a timelike geodesic of AdS 3 ; (2) The two spacelike planes intersect in AdS 3 , whereas the two timelike planes intersect in two spacelike geodesics of AdS 3 .
Proposition 3.10 follows from the more general Proposition 3.12 below, which also includes the collapsed case. We will consider only the degeneration of cusp groups to a collapsed cusp group when the two planes which coincide are spacelike, as in the following definition: Definition 3.11 (Collapsed cusp group for AdS 3 ). The image of a representation of Γ rect into Isom(AdS 3 ) is called a collapsed cusp group if the four generators are sent to reflections along three distinct planes, two timelike and one spacelike, asymptotic to a common point at infinity.
Proposition 3.12. Let ρ : Γ rect → Isom(AdS 3 ) be a representation whose image is a cusp group or a collapsed cusp group. For all nearby representations ρ , exactly one of the following possibilities holds: (1) If s 1 and s 2 are generators such that ρ(s 1 ) = ρ(s 2 ) is a reflection in a spacelike plane, then ρ (s 1 ) = ρ (s 2 ).
(2) The image of ρ is a cusp group.
(3) The ideal closures of the two spacelike planes are disjoint in AdS 3 , whereas the two timelike planes intersect in a timelike geodesic of AdS 3 . (4) The two spacelike planes intersect in AdS 3 , whereas the two timelike planes intersect in two spacelike geodesics of AdS 3 .
Proof. By Lemmas 3.3 and 3.4, it is sufficient to analyse a neighborhood of a lift f : S → R 4 of ρ, where S is the standard generating set of Γ rect . Let us denote s 1 , s 2 the generators which are sent by ρ to a reflection in a spacelike plane, and t 1 , t 2 those sent to a reflection in a timelike plane. The same will occur for representations nearby ρ. Let us fix a nearby representation ρ and a lift f : Suppose that X 1 = ±X 2 , for otherwise we are in the case of item (1). Up to the action of O(q −1 ) (see Remark 3.5) and up to changing signs, we can assume once and forever that Suppose first that the hyperplanes H Y2 , H X1 and H Y1 are asymptotic to the same point at infinity p. (As a side remark, observe that in this case they are also asymptotic to the antipodal point −p.) We can assume p = (0, 0, 1, 1) ∈ ∂AdS 3 . Together with the orthogonality of Y 2 with X 1 , this implies (up to changing the sign if necessary) for some parameter a = 0. Applying the orthogonality of X 2 with Y 1 and Y 2 , we now find (always up to a sign) for some b, which implies that H X2 also is asymptotic to the point p = (0, 0, 1, 1). Thus we still have a cusp group and we are in the case of item (2).
. There are two possibilities: either 1 and 2 intersect in H 0 X1 , or they are ultraparallel. See Figure 4 and the related Figure 3.
where θ is the angle between the two geodesics in H 0 X1 . In this case, the two timelike planes H Y1 and H Y2 have timelike intersection by Lemma 2.6 (the intersection is indeed the timelike geodesic (cos(s), 0, 0, sin(s))). Imposing the orthogonality of H X2 with H Y1 and H Y2 , we find (up to a sign) X 2 = (cos ϕ, 0, 0, sin ϕ) , which means that H X1 and H X2 are disjoint in AdS 3 by Lemma 2.5. (The parameter ϕ is indeed the timelike distance between H X1 and H X2 , which is achieved on the timelike geodesic we have just introduced.) So, in this case item (3) of the statement holds. If 1 and 2 are ultraparallel, we can assume where θ is now the distance between the two aforementioned geodesics in H 0 X1 . In this case, H Y1 and H Y2 have spacelike intersection (which is the geodesic (cosh(s), 0, sinh(s), 0), see Lemma 2.6). Imposing again the orthogonality of H X2 with H Y1 and H Y2 , and changing sign if necessary, we find namely, H X1 and H X2 intersect in AdS 3 by Lemma 2.5 (the parameter ϕ now being their angle of intersection). Thus, item (4) of the statement holds. This concludes the proof.
Remark 3.13. In the proof of Proposition 3.12 we have used only that ρ can be continuously deformed to ρ . Hence the conclusions of Proposition 3.12 and Proposition 3.9 actually hold on the entire connected component of Hom refl (Γ rect , G) containing ρ.

Rigidity in dimension four.
Let us now move to dimension four. Let Γ cube be the group generated by the reflections in the faces of a Euclidean cube. The group Γ cube has 6 generators, one for each face, and 12 commutation relations, one for each edge of the cube, involving the two faces adjacent to that edge. Of course, there is also a square-type relation for each generator. There is no relation between the generators corresponding to opposite faces.
Definition 3.14 (Cusp group in dimension 4). The image of a representation of Γ cube into Isom(AdS 4 ) or Isom(H 4 ) is called a cusp group if the 6 generators are sent to reflections in 6 distinct hyperplanes asymptotic to a common point at infinity.
In the AdS case, among these 6 hyperplanes, two opposite hyperplanes are necessarily spacelike, while the other 4 are timelike.
The following proposition is the fundamental property that can be roughly rephrased as: "cusp groups stay cusp groups". Its hyperbolic counterpart is proved in [KS10, Lemma 5.3].
Proposition 3.15. Let ρ : Γ cube → Isom(AdS 4 ) be a representation whose image is a cusp group. Then all nearby representations are cusp groups.
Similarly to dimension three, we will obtain Proposition 3.15 as a special case of a more general statement including the collapsed case. Let us first give the definition of collapsed cusp group, where two non-commuting generators can be sent to the same reflection (along a spacelike hyperplane in the AdS case): Definition 3.16 (Collapsed cusp group in dimension 4). The image of a representation of Γ cube into Isom(H 4 ) or Isom(AdS 4 ) is called a collapsed cusp group if the 6 generators are sent to reflections along 5 distinct hyperplanes asymptotic to a common point at infinity. In the AdS case, we require that the unique reflection associated to two generators is along a spacelike hyperplane.
Let us now formulate and prove the more general version of Proposition 3.15.
Proposition 3.17. Let ρ : Γ cube → Isom(AdS 4 ) be a representation whose image is a cusp group or a collapsed cusp group. For all nearby representations ρ , exactly one of the following possibilities holds: (1) If s 1 and s 2 are generators such that ρ(s 1 ) = ρ(s 2 ) is a reflection in a spacelike hyperplane, then ρ (s 1 ) = ρ (s 2 ).
(2) The image of ρ is a cusp group.
Proof. Similarly to the three-dimensional case treated in the previous section, any representation ρ nearby ρ lies in Hom refl (Γ cube , G), hence it sends the 6 standard generators of Γ cube to reflections. Moreover, the hyperplanes of ρ have the same type (spacelike or timelike) as for ρ.
Let us pick a lift f : S → R 5 of ρ , for S the standard generating set of Γ cube . Denote by s 1 , s 2 the two generators corresponding to opposite faces of the cube which are sent to reflections in spacelike hyperplanes, and X i = f (s i ) (so that q −1 (X i ) = −1). Similarly we define Y i and Z i for i = 1, 2, on which q −1 takes value 1. Hence each of this 6 vectors is orthogonal to 4 of the others: more precisely, A i is orthogonal to all the others except A j , for A ∈ {X, Y, Z} and i, j = 1, 2. Now, let us assume that the hyperplanes H X1 and H X2 do not coincide, that is X 1 = ±X 2 . We shall show that the image of ρ is still a cusp group.
Let us start by considering the intersection with a connected component H 0 , whose associated reflections give a representation Γ rect → Isom(H 3 ) which is nearby a (rectangular) cusp group. As in the proof of Proposition 3.10, it is easy to see that if this representation of Γ rect is a cusp group in H 0 X1 , then necessarily also H X2 is asymptotic to a common point at infinity with H Y1 , H Y2 , H Z1 , H Z2 . Therefore the image of Γ cube is still a cusp group, since we are assuming that H X2 = H X1 .
Hence let us assume that the representation of Γ rect is not a cusp group, and we will derive a contradiction. By Proposition 3.8 (up to relabelling) we may assume that . This implies that H Y1 ∩ H Y2 is a timelike plane (i.e. a copy of AdS 2 ), while H Z1 ∩ H Z2 is spacelike (i.e. a copy of H 2 ). To see this, one can in fact assume that, up to the signs, and apply Lemma 2.6 -and similarly for Z 1 and Z 2 . Now, let us consider the intersection with H Y1 , which is a copy of AdS 3 . We have thus a representation of Γ rect acting on this copy of AdS 3 as a cusp group or collapsed cusp group, with generators which are reflections in is also spacelike, and therefore we are in the situation of Proposition 3.12 item (4), recalling that H X1 = H X2 by our assumption.
On the other hand, considering the intersection with H Z1 , which is again a copy of AdS 3 , since H Y1 ∩H Y2 is timelike, we find that H Y1 ∩H Y2 ∩H Z1 is a timelike geodesic. By Proposition 3.12 item (3), H X1 ∩H Z1 and H X2 ∩H Z1 do not intersect in H Z1 , which in turn implies (since H X1 and H X2 are both orthogonal to H Z1 ) that H X1 and H X2 are disjoint in AdS 4 . This contradicts the conclusion of the previous paragraph.
Remark 3.18. In case (1) of Proposition 3.17, i.e. when ρ(s 1 ) = ρ(s 2 ), the following possibility is not excluded: for some deformation ρ of ρ, the remaining 4 generators s 3 , . . . , s 6 (which are sent by ρ to a rectangular cusp group in a copy of H 3 ) are not sent by ρ to a cusp group.
The analogous property for H 4 , which is a generalisation of [KS10, Lemma 5.3] can be proved along the same lines. We state it here: Proposition 3.19. Let ρ : Γ cube → Isom(H 4 ) be a representation whose image is a cusp group or a collapsed cusp group. For all nearby representations ρ exactly one of the following possibilities holds: (1) If s 1 and s 2 are generators such that ρ(s 1 ) = ρ(s 2 ), then ρ (s 1 ) = ρ (s 2 ).
(2) The image of ρ is a cusp group.

The hyperbolic and AdS character variety of Γ 22
In this section, we study the Isom(H 4 ) and Isom(AdS 4 ) character varieties of the group Γ 22 near the conjugacy classes of the holonomy representations ρ t found in [KS10,RS]. We prove here the "topological part" of Theorem 1.1 (Theorem 4.16) in the hyperbolic and AdS case.
as the group generated by the hyperbolic reflections along the hyperplanes determined by the 22 vectors in Table 1. These hyperplanes bound a right-angled polytope in H 4 of infinite volume, which is obtained by "removing two opposite walls" from the ideal right-angled 24cell.
All the dihedral angles between two intersecting hyperplanes are right. Therefore Γ 22 is a right-angled Coxeter group. We will consider Γ 22 as an abstract group, that is the right-angled Coxeter groups on 22 generators satisfying the following relations: • s 2 = 1 for each generator s, • s 1 s 2 = s 2 s 1 for each pair s 1 , s 2 of generators such that the corresponding vectors in Table 1 are orthogonal with respect to the bilinear form b 1 .
The reader can check from Table 1 that there are no commutation condition between two generators of the same type, that every i + commutes with 4 vectors of type j − (including i − ), and every X ∈ {A, . . . , F } commutes with i − and i + for 4 choices of i ∈ {0, . . . , 7}. Hence there are 8 · 4 + 6 · 8 = 80 commutation relations. Altogether, there are 102 = 22 + 80 relations.

4.2.
A curve of geometric representations. Let us now introduce the representations of our interest, which appear in the statement of Theorem 1.1. Unlike the introduction, we will omit the superscript G hereafter, and the ambient geometry we consider will be clear from the context. Definition 4.1 (The two paths ρ t ). For t ∈ (−1, 1), we define ρ t to be the representation of Γ 22 in Isom(H 4 ) (resp. Isom(AdS 4 )) sending each generator s of Γ 22 to the hyperbolic (resp. AdS) reflection r ft(s) associated to the corresponding vector f t (s) of Table 2 (resp. Table 3).
Some comments are necessary to explain Definition 4.1 and the tables involved: (1) It can be checked that all the orthogonality relations (with respect to the bilinear form b 1 ) between vectors in Table 1 are maintained for the vectors in Table 2 with respect to b 1 , and in Table 3 Table 2 coincide with those of Table 1 for t = 1. Hence, in the hyperbolic case, the path of representations ρ t is a deformation of the reflection group of the aforementioned right-angled polytope with 22 facets. For t ∈ (0, 1), this coincides with the path of representations exhibited in [KS10]. For t ∈ (−1, 0), the representation ρ t is obtained by conjugating ρ −t by the reflection r in the "horizontal" hyperplane x 4 = 0 (This is seen immediately using Remark 3.5.) (4) On the Anti-de Sitter side, the path ρ t has been exhibited in [RS] for t ∈ (−1, 0).
Again, the path is extended here for positive times by conjugation by r.   Table 4. The vectors v i , v X ∈ R 1,3 defining the bounding planes of an ideal rightangled cuboctahedron in H 3 . These vectors are involved also in the Definition 6.3, introducting the cocycles τ λ in the vector space Z 1 0 (Γ 22 , R 1,3 ).
(5) Both these paths occur as the holonomy representations of a deformation of hyperbolic and Anti-de Sitter cone-orbifold structures. The purpose of our previous work [RS] was to describe the geometric transition from hyperbolic (t > 0) to Anti-de Sitter (t < 0) structures. Since here we are interested in the Isom(H 4 )-and Isom(AdS 4 )chatacter varieties on their own, we found more useful to treat the two paths ρ t separately, and extend each of them by conjugationwith the orientation-reversing transformation r also for negative (resp. positive) times.
4.3. The collapsed representation and the cuboctahedron. For t = 0 the hyperbolic and Anti-de Sitter representations ρ 0 take value in the stabiliser of the hyperplane given by {x 4 = 0}. Unlike the case t = 0, these representations are not holonomies of hyperbolic/AdS orbifold structures, but correspond to what we call the collapse of the respective geometric structures. Let us consider Isom(H 4 ) and Isom(AdS 4 ) as subgroups of GL(5, R). Then the representations ρ 0 agree for the hyperbolic and AdS case. Indeed, defining consists of matrices in the block form: The stabiliser G 0 is isomorphic to Isom(H 3 ) × Z/2Z, where the Z/2Z-factor is generated by the reflection r of Equation (5), which acts by switching the two sides of {x 4 = 0}. Under this isomorphism, the representation ρ 0 reads as:  Table 4, define the bounding planes H v i and H v X of an ideal right-angled cuboctahedron in H 3 . The triangular faces of this cuboctahedron are of type i, while the quadrilateral faces are of type X (see Figure 5).
4.4. The conjugacy action. We are ready to start the study of the hyperbolic and AdS character varieties of Γ 22 near the representation ρ t introduced in Definition 4.1. We begin by analysing the action of G by conjugation on Hom(Γ 22 , G). For t = 0, nearby ρ t the action of G is "good", namely is free and proper, as we will see in the following two lemmas.
Lemma 4.2. For t = 0, the stabiliser of ρ t in G is trivial. The stabiliser of ρ 0 in G is the order-two subgroup generated by the reflection r in the hyperplane {x 4 = 0}.
Proof. We give the proof for the hyperbolic and AdS case at the same time, since they are completely analogous. By Remark 3.5, any element in the stabiliser of ρ t is induced by a matrix A ∈ O(q ±1 ) which maps every vector f t (s) in Table 2 or Table 3 either to itself or to its opposite. Since the 6 vectors f t (A), . . . , f t (F ) do not depend on t and generate the hyperplane {x 4 = 0}, the matrix A must preserve the hyperplane {x 4 = 0}.
Moreover, let P t be the polytope bounded by the 22 hyperplanes orthogonal to the vectors of Tables 2 or 3. It was proved in [MR18, Proposition 3.19] and [RS,Proposition 7.21] that the intersection of P t with the hyperplane defined by the equation x 4 = 0 is constant and is an ideal right-angled cuboctahedron in H 3 (see Section 4.3). Since the action of A on H 3 necessarily preserves each face of the cuboctahedron, it follows that A must act on the linear hyperplane {x 4 = 0} as ±id.
This shows that the only non-trivial candidates for A are ±r, where r is the reflection of Equation (5). For t = 0, the reflection r preserves all the hyperplanes orthogonal to the vectors of Tables 2 or 3, hence the associated element in G generates the stabiliser of ρ 0 . When t = 0, the reflection r does not preserve any of the hyperplanes of the form H ft(i + ) and H ft(i − ) , hence the stabiliser of ρ t is trivial in this case.
The next lemma will be useful to show that the action of G + by conjugation is proper, in a suitable region of Hom refl (Γ 22 , G).
Lemma 4.3. Suppose that η n is a sequence in Hom refl (Γ 22 , G) converging to some ρ t , and h n is a sequence in G such that h n · η n converges. Then h n has a subsequence that converges in G.
Proof. Suppose that η n → ρ t and h n is a sequence in G such that h n · η n → η ∞ . Since Hom refl (Γ 22 , G) is clopen in the representation variety, the limit point η ∞ is in Hom refl (Γ 22 , G). Passing to the finite cover g −1 (0) of Lemma 3.4, and up to taking subsequences, we can then assume to have a sequence f n in g −1 (0) (projecting to η n ) such that f n → f ∞ and h n · f n → f ∞ . Here we are thinking of f n , f ∞ , f ∞ as functions from the standard generators of Γ 22 to R 5 , and (by a small abuse of notation) h n is a sequence in O(q ±1 ) acting by the obvious action on R 5 (see Remark 3.5).
We have to show that h n converges in O(q ±1 ) up to subsequences. Recall that f ∞ is a lift in g −1 (0) of ρ t , and therefore (up to changes of sign) the vectors f ∞ (s) are given by Table 2 or Table 3  The linear isometry h n ∈ O(q ±1 ), considered as a 5-by-5 matrix, is therefore determined by the condition that h n sends the basis {f n (s 1 ), . . . , f n (s 5 )} to {h n · f n (s 1 ), . . . , h n · f n (s 5 )}. More concretely, we can write h n (as a matrix) as (h n,1 ) −1 • h n,2 , where h n,1 is the matrix sending the standard basis to the basis {f n (s 1 ), . . . , f n (s 5 )}, and h n,2 is the matrix sending the standard basis to the basis {h n ·f n (s 1 ), . . . , h n ·f n (s 5 )}. Since f n and h n ·f n are converging sequences, we have that h n,1 → h ∞,1 and h n,2 → h ∞,2 , and moreover h ∞,1 is invertible since f ∞ (s 1 ), . . . , f ∞ (s 5 ) is a basis.
Therefore h n converges to a 5-by- is closed in the space of 5-by-5 matrices. This concludes the proof.
The two lemmas have some consequences on the character variety, which we now define: Definition 4.4 (Character variety). Let G be Isom(H n ), G HP n or Isom(AdS n ), and G + denote its subgroup of orientation-preserving transformations. Given a finitely generated group Γ, we call character variety of Γ in G the quotient X(Γ, G) = Hom(Γ, G)/G + by the action of G + by conjugation.
Remark 4.5. It follows from Lemmas 4.2 and 4.3 that the G + -action is locally a product in a neighbourhood of ρ t . More precisely, there is a neighbourhood of the G + -orbit of ρ t in Moreover the action of G + corresponds, under this homeomorphism, to left multiplication on the second factor.
Remark 4.6. Let us suppose G is Isom(H n ) or Isom(AdS n ); in other words G is reductive. Thanks to some well-known results from GIT (see for instance the concise exposition in [CLM18, Section 2] and the references therein), the GIT quotient Hom(Γ 22 , G)/ /G + can be identified with the "Hausdorff quotient" of the representation variety by conjugation: namely, the quotient by G + of the subset of Hom(Γ, G) consisting of points with closed G + -orbits.
In the portion of the character variety of our interest, no non-Hausdorff pathological situation arises. More precisely, the GIT quotient Hom(Γ 22 , G)/ /G + coincides with the ordinary topological quotient in a neighbourhood of each [ρ t ]. (This holds similarly for Hom(Γ 22 , G)/ /G.) For, it follows from Lemma 4.3 that: • The G + -action is proper on G + · {ρ t | t ∈ (−1, 1)}.
• For each t, the G + -orbit of ρ t is closed. (This follows by applying Lemma 4.3 to the constant sequence η n ≡ ρ t ).
Actually the latter is true in a neighborhood of {ρ t | t ∈ (−1, 1)}, since in the proof of Lemma 4.3 we only used that, for five generators s 1 , . . . , s 5 of Γ 22 , the corresponding vectors in R 5 are linearly independent, and this is still true in an open neighborhood.
In fact, our argument shows a little more, namely that if ρ is in such a neighbourhood, then [ρ] is separated from any other point in X(Γ 22 , G). This is because, if [ρ] were not separated from [ρ ], we would have a sequence ρ n → ρ and a sequence h n such that h n ρ n h −1 n converges to ρ . But Lemma 4.3 shows that h n → h ∞ up to subsequences, hence by continuity h ∞ conjugates ρ and ρ , namely Proposition 4.7. For t ∈ (0, 1), the set Hom(Γ 22 , Isom(H 4 )) is a smooth 11-dimensional manifold near ρ t .
(Recalling that for negative times ρ t is a conjugate of ρ −t , the result holds for t ∈ (−1, 0) as well.) Our main purpose is to extend and generalise the analysis for t = 0, and do similarly for the Isom(AdS 4 )-representation variety.
Let us first briefly sketch the lines of the proof Proposition 4.7 given in [KS10]. By Lemma 3.4 (recall g : R (n+1)|S| → R |S|+|R| from the proof), it suffices to show that g −1 (0) is a smooth submanifold of R 110 near any preimage of ρ t0 , for all t 0 ∈ (0, 1). Let be as in Table 2, so giving an embedding of (0, 1) into g −1 (0) ⊂ R 110 going through a preimage of ρ t0 . The proof in [KS10] essentially consists in showing that the kernel of g : R 110 → R 102 is 11-dimensional for t ∈ (0, 1). Since there is a 10-dimensional smooth orbit given by the action of Isom + (H 4 ), the proof boils down to showing that the tangent space to the orbit has a 1-dimensional complement, which is indeed given by the tangent space to the 1-dimensional submanifold {f t | t ∈ (0, 1)}.
Since the action of Isom + (H 4 ) is smooth, it then follows that the Isom + (H 4 )-orbit of the curve {ρ t | t ∈ (0, 1)} is a smooth 11-dimensional manifold, on which the Isom + (H 4 )action by conjugation is free and proper by Lemma 4.2 and Lemma 4.3. Hence it follows from Proposition 4.7 that X(Γ 22 , Isom(H 4 )) is a 1-dimensional smooth manifold near [ρ t ], for t ∈ (0, 1).
In the next sections, we will prove the analogous of Proposition 4.7 for the AdS case. However, we are interested also in the study of the character variety near "the collapse", that is the point [ρ 0 ]. Hence we will prove a more detailed statement.
Definition 4.8 (The set Hom 0 ). We define Hom 0 (Γ 22 , G) as the subset of Hom refl (Γ 22 , G) of representations ρ such that the following holds. Let s 1 , s 2 be any pair of generators of Γ 22 such that the hyperplanes fixed by ρ t (s 1 ) and ρ t (s 2 ) are either asymptotic or equal for some t = 0. Then, so are the hyperplanes fixed by ρ(s 1 ) and ρ(s 2 ).
Recall from Lemmas 2.4 and 2.5 that two hyperplanes are asymptotic or equal if and only if, using the bilinear form b 1 for H 4 and b −1 for AdS 4 , the product of their orthogonal unit vectors is 1 in absolute value. It is thus easy to check from Tables 2 and 3 that this condition is preserved by the deformation f t for all t both in the hyperbolic and AdS case, and thus the definition is well-posed (i.e. it does not depend on the choice of t = 0).
In the setting of Lemma 3.4, Hom 0 (Γ 22 , G) corresponds to a subset of g −1 (0) ⊂ R 110 defined by the vanishing of 36 more quadratic conditions. Indeed, for each of the 12 ideal vertices of the polytope P t bounded by the hyperplanes of Tables 2 and 3, we have 3 asymptoticity conditions (see [RS,Proposition 7.13]). Hence Hom 0 (Γ 22 , G) is locally homeomorphic to the zero locus of a function g 0 : R 110 → R 138 extending g. More precisely: Lemma 4.9. The set Hom 0 (Γ 22 , G) is finitely covered by a disjoint union of subsets of R 110 defined by the vanishing of 138 quadratic conditions. Remark 4.10. For simplicity of exposition, from now on we will work in the AdS setting, i.e. in the case G = Isom(AdS 4 ). All what follows can be easily adapted to the hyperbolic case. We will therefore omit the proofs and only highlight the points where differences with respect to the AdS case occur.
The essential property we will prove is that near each of the representations ρ t the variety Hom 0 (Γ 22 , G) is smooth. Hence the goal of the next two sections is to prove the following: Proposition 4.11. For t ∈ (−1, 1), the set Hom 0 (Γ 22 , Isom(AdS 4 )) is a smooth 11-dimensional manifold near ρ t .
The proof of Proposition 4.11 will be given at the end of Section 4.7. From the results on cusp rigidity established in Section 3.3, we obtain the smoothness of Hom(Γ 22 , Isom(AdS 4 )) for t = 0 as a direct corollary: Proof. It is not difficult to check that, when t = 0, for every pair of generators s 1 , s 2 of Γ 22 such that the associated hyperplanes H ft(s1) and H ft(s2) are asymptotic, there are 4 other generators s 3 , . . . , s 6 such that the reflections r s1 , . . . , r s6 generate a cusp group in Isom(AdS 4 ). By Lemma 3.15, the asymptoticity conditions are preserved since cusp groups stay cusp groups under small deformations. Hence a neighbourhood of ρ t in Hom(Γ 22 , Isom(AdS 4 )) is actually contained in Hom 0 (Γ 22 , Isom(AdS 4 )). The proof now follows from Proposition 4.11.
The next sections will be devoted to the proof of Proposition 4.11. We will adapt some of the ideas of [KS10, Sections 5, 11, 12] used in the proof of Proposition 4.7 in the hyperbolic case. An analogous argument shows that the statement of Proposition 4.11 holds also for the H 4 -character variety, which for t = 0 is new with respect to the results of [KS10]. 4.6. Infinitesimal deformations of the letter generators. Recall Lemma 4.9. Throughout this and the following sections, we denote by the quadratic function defining the clopen subset of Hom 0 (Γ 22 , Isom(AdS 4 )) that contains the lifts of the representations ρ t . A continuous lift of the path t → ρ t is defined by f t in Table 3. To prove Proposition 4.11 in the AdS case, it then suffices to show that for all t ∈ (−1, 1) the set g −1 0 (0) ⊂ R 110 is a smooth 11-dimensional manifold near each f t .
Notation. Let us fix t ∈ (−1, 1). For simplicity, by an abuse of notation, in this and next section we denote f t (s) ∈ R 5 by s. In other words, in what follows s ∈ R 5 denotes a vector (of q −1 -norm 1 or −1 depending whether the corresponding hyperplane in AdS 4 is timelike or spacelike, respectively) from Table 3, and is therefore implicitly considered as a function of t. Its derivative in t will be denoted byṡ. The symbol ({i + }, {j − }, {X}) will denote the corresponding element of g −1 0 (0) ⊂ R 110 , as a function from the standard generators of Γ 22 to R 5 , while ({i + }, {j − }, {Ẋ}) will denote an element in the kernel of the differential of g 0 at ({i + }, {j − }, {X}), and will be called an infinitesimal deformation of ({i + }, {j − }, {X}).
Observe that the vectors A, . . . , F of Table 3 are constant in t, hence the derivative of the path in g −1 0 (0) provided by Table 3 The first step in the proof of Proposition 4.11 is to show that, up to this infinitesimal action, we can assume that any infinitesimal deformation vanishes at least on 4 elements of {A, . . . , F }. Up to the action of a ∈ so(q −1 ) as in (8), we can assume thaṫ and thatĖ = (0, 0, 0, 0, ) andḞ = (0, 0, 0, 0, φ) (10) for some , φ ∈ R.
The analogous lemma in the hyperbolic case, for t = 0, has been proved in [KS10, Proposition 11.1], and in fact the arguments here follow roughly the same lines as their proof. However, the first part of their proof uses a nice geometric argument which would be complicated to adapt to AdS geometry. For this reason, we rather use a linear algebra argument here.
Notation. To simplify the notation, from here to the end of Section 4.7, we denote by ·, · the bilinear form b −1 . If one wants to repeat the proof for G = Isom(H 4 ), then ·, · should denote b 1 . The reader should pay attention that in Section 6 the bracket ·, · will instead be used to denote the Minkowski bilinear form on R 4 .
Proof. The proof will follow from three claims.
To show the first claim, consider the basis {A, B, C, D, e 4 } of R 5 , where e 4 = (0, 0, 0, 0, 1). Recall that matrices a in the Lie algebra so(q −1 ) are characterised by the condition that a · u, w + u, a · w = 0 (12) for every u, w, and that it suffices in fact to check the condition for all pair of elements u, w of our fixed basis. Moreover, to define the matrix a in so(q −1 ), it suffices to define it on 4 vectors of the basis of R 5 , such that (12) holds when u, w are chosen among these 4 vectors.
The definition of a on the last vector of the basis is then uniquely determined by (12). Let us now apply these preliminary remarks. By differentiating the conditions Equations (13) and (14) show that any linear transformation a ∈ so(q −1 ) sending A toȦ, B toḂ and C toĊ satisfies the conditions of (12) for all pairs of u, w chosen in {A, B, C}.
It remains to define a on the remaining two elements D and e 4 of the basis. Equation (12) imposes the value of a · D, u and a · e 4 , u for all u ∈ {A, B, C}. Moreover we must have a·D, D = a·e 4 , e 4 = 0. Therefore a·D and a·e 4 can be chosen with one degree of freedom given by the value of a · D, e 4 = − D, a · e 4 . This shows that we can find a ∈ so(q −1 ) satisfying Equation (11), and our first claim is proved. Second, we claim that we can further assume that Ḋ , e 4 = 0 .
To see this second claim, by repeating the same reasoning as in the beginning of this proof, it suffices to find another a ∈ so(q −1 ) so that a · A = a · B = a · C = 0 and a · D = Ḋ , e 4 e 4 .
As observed in the proof of Corollary 4.12, since A, D = −1, the vectors A and D play the role of two non-commuting generators (reflections along two timelike hyperplanes that are asymptotic) of a cusp group generated by the images of A, D, 3 + , 3 − , 2 + , and 2 − . By the assumption that asymptoticity conditions are preserved (recall that we are in Hom 0 ), any deformation of A and D satisfies A, D = −1. So, by differentiating and usingȦ = 0, we obtain A,Ḋ = 0. Analogously, B,Ḋ = 0.

4.7.
Infinitesimal deformations of the positive and negative generators. We conclude in this section the proof of Proposition 4.11. A direct computation from Table 3 shows that the tangent vector to our explicit path f t in g −1 0 (0) is given by: for some λ ∈ R (depending on t).
This provides the conclusion of the proof of Proposition 4.11.
Proof of Proposition 4.11. Let us fix t ∈ (−1, 1). We now show that the kernel of the differential of g 0 : R 110 → R 138 is 11-dimensional at ({i + }, {j − }, {X}) ∈ g −1 0 (0). Lemmas 4.13, 4.14 and 4.15 showed that every element in the kernel of dg 0 is of the form (16) up to adding an element of the form (8), that is an element tangent to the orbit of the Isom(AdS 4 )-action. It is also easy to see that such element in the tangent space of the orbit is unique, for if two elements a 1 and a 2 have this property, it follows that a := a 1 − a 2 satisfies a · X = 0 for X = A, B, C, D and the characterising conditions (12) (already used in Lemma 4.13) show that a = 0. The very same argument shows that the map defined in (8) from the Lie algebra isom(AdS 4 ) into the kernel of the differential of g 0 (whose image is the tangent space to the orbit of the Isom(AdS 4 )-action) is injective.
In other words, the 10-dimensional tangent space of the orbit has a 1-dimensional complement, consisting precisely of the elements of the form (16), hence the kernel of the differential of g 0 has dimension 11. By the constant rank theorem, g −1 0 (0) is a manifold of dimension 11 near the elements in the orbit of ρ t . 4.8. Topology of the neighbourhood U. We now state a weaker version of Theorem 1.1: • V corresponds to the x 13 -axis, and consists of the conjugacy classes of the holonomy representations ρ G t ; • H corresponds to {x 13 = 0}, identified to a neighbourhood of the complete hyperbolic orbifold structure of the ideal right-angled cuboctahedron in its deformation space.
The group G/G + ∼ = Z/2Z acts on S by changing sign to the last coordinate x 13 .
This statement is weaker than Theorem 1.1 because it gives a purely topological description of U, while the smoothness and transversality of its components will be proved in Section 7.
We prove here Theorem 4.16 in the Anti-de Sitter case. The proof in the hyperbolic case is completely analogous, so we omit it, while the proof in the HP case will be given later in Section 6.4. We decided to give a proof only in the AdS case, since the fact that the points [ρ t ] for t > 0 form a smooth curve (Proposition 4.7) has already been proved in [KS10], while its AdS counterpart is completely new. The proof for the hyperbolic case is analogous (recall Remark 4.10). Moreover, the description of the collapse (namely, at the representation ρ 0 ) is also new in both (hyperbolic and AdS) cases.
Proof of Theorem 4.16 -AdS case. We split the proof into several steps.
Since by Lemma 4.3 the Isom + (AdS 4 )-action by conjugation is free on {ρ t } t∈(−1,1) , the map (g, t) → g · ρ t defines a continuous injection where by Proposition 4.11 the latter is a smooth 11-dimensional manifold. By the invariance of domain, this injection is a homeomorphism onto its image, which is V. By Lemma 4.2 and Lemma 4.3, the Isom + (AdS 4 )-action by conjugation is free and proper on V thus the projection in the quotient X(Γ 22 , Isom(AdS 4 )) is which is homeomorphic to a line.
Step 2 : The second component H is defined as follows.
Recall from Section 4.3 that we have a fixed copy H 3 ⊂ AdS 4 defined by x 4 = 0 and x 0 > 0, fixed by the reflection r. Its stabiliser G 0 is Isom(H 3 ) × r , where we consider Isom(H 3 ) as a subgroup of Isom(AdS 4 ).
Consider the reflection group Γ co of the ideal right-angled cuboctahedron. We define the Indeed, (1) holds because, using that r commutes with the elements of Isom(H 3 ) < Isom(AdS 4 ), the images of the generators in Isom(AdS 4 ) through Ψ η satisfy the relations of Γ 22 , so that Ψ η is indeed a representation of Γ 22 in Isom(AdS 4 ). The equivariance of Ψ is clear using again that r commutes with Isom(H 3 ). It also follows that Ψ is well defined. Inded, by the equivariance of Ψ, if η 1 and η 2 are conjugate in Isom(H 3 ) then Ψ η1 and Ψ η1 are conjugate in Isom(AdS 4 ). Up to composing with r, which commutes with both Ψ η1 and Ψ η2 , then the latter are conjugate in Isom + (AdS 4 ).
The setX(Γ co , Isom(H 3 )) is a 12-dimensional manifold in a neighborhood (say H 0 ) of [η 0 ], since it corresponds to a neighbourhood of the complete hyperbolic orbifold structure of the right-angled cuboctahedron in its deformation space. To show this, the same proofs of [KS10, Proposition 5.2] apply (see also the related discussion in [KS10, Section 5]), as a well-known "reflective" orbifold version of Thurston's hyperbolic Dehn filling [Thu79] (note that the ideal cuboctahedron has 12 cusps).
Therefore a neighborhood H 0 of [η 0 ]X(Γ co , Isom(H 3 )) is homeomorphic to R 12 , and we can also assume that Ψ| H0 is a homeomorphism onto its image. Then let us define H := Ψ(H 0 ).
Step 4 : Let us now show that the point [ρ 0 ] ∈ X(Γ 22 , Isom(AdS 4 )) has a neighbourhood U which is contained in the union of the two components V and H.
To see this, let ρ be a representation nearby ρ 0 . We claim that if two generators which are sent by ρ 0 to the same reflection r (hence necessarily of the form i + and j + ) are sent to reflections in coinciding hyperplanes also by ρ, then all generators 0 + , . . . , 7 + are sent by ρ to the same reflection. That is, if ρ(i + ) = ρ(j + ) for some i, j, then ρ(i + ) = ρ(j + ) for all i, j. This will show our thesis by the rigidity property of Proposition 3.17: if [ρ] is not on the "horizontal" component H, then no two letter generators are sent to the same reflection, and thus all the collapsed cusp groups of ρ 0 are cusp groups for ρ. That is, ρ lies in Hom 0 (Γ 22 , Isom(AdS 4 )) and thus in the "vertical" component V, since V is open in Hom 0 (Γ 22 , Isom(AdS 4 )).
To prove the claim, suppose that two generators i + and j + are such that ρ(i + ) = ρ(j + ). By the symmetries of the polytope P t (see [RS,Lemma 7.6]) and Proposition 3.17, we can assume the two generators are 0 + and 1 + . Up to conjugation in Isom(AdS 4 ), we can also assume ρ(0 + ) = ρ(1 + ) = r. To simplify the notation, let f be a preimage of ρ in g −1 (0), which associates to each generator of Γ 22 a vector in R 5 of square norm 1 or −1 with respect to q −1 .
For a small deformation of ρ 0 , the vectors f (1 − ), f (2 − ), f (3 − ) and f (A) are linearly independent, because they are for ρ 0 (see Table 3). Hence the conditions of being orthogonal to these 4 vectors define a linear system of 4 independent equations, which are satisfied by e 4 . Hence f (2 + ), which is a solution of the system, coincides with e 4 up to rescaling. Since q(f (2 + )) = −1, we can assume that f (2 + ) = e 4 . Namely, ρ(2 + ) = r. By arguing similarly for 3 + and then for all the other generators, one easily finds sufficiently many relations to show that ρ(i + ) = r for each generator i + ∈ {0 + , . . . , 7 + }, and therefore ρ is in H. This proves the claim.
Step 5 : Summarising the previous steps, we have shown that the class [ρ 0 ] has a neighborhood U which only consists of points of H and V. Since we already know that H and V are smooth manifolds outside of ρ 0 , it is harmless to enlarge U so that it contains entirely H and V.
We have therefore obtained a neighborhood U of where the two components are precisely H and V.
Step 6 : It remains to prove the last sentence about the action of the group generated by the coset of the reflection r. This is now simple: on the one hand, as observed after Definition 4.1, conjugation by r acts on V, which is homeomorphic to (−1, 1), by [ρ t ] → [ρ −t ]. On the other hand, by construction of H, conjugation by r fixes pointwise the elements in H, which are of the form Ψ η for some η : Γ co → Isom(H 3 ). This concludes the proof.
We conclude the section with a couple of observations on the nature of the fixed points for the action of G on Hom(Γ 22 , G).
Lemma 4.2 shows that the stabiliser of each point ρ t in Hom(Γ 22 , G), for the conjugacy action of G, is trivial, except ρ 0 which has stabiliser r . In fact, a small adaptation of the proof shows that, in a neighborhood of ρ 0 , the stabiliser of all points in the the horizontal component H is as well the group Z/2Z generated by r. This is because we can find a neighborhood of ρ is in the image of Ψ such that, for a lift f of ρ, the vectors f In conclusion, let us consider the full quotient Hom(Γ 22 , G)/G, which is a Z/2Z-quotient of X(Γ 22 , G), where Z/2Z ∼ = G/G + . A local picture of this full quotient is given in Figure  1 (right), as a consequence of the fact that the generator of Z/2Z acts by changing sign to the x 13 -coordinate, hence as a "reflection" with respect to the horizontal component H. The "horizontal" component (which is the projection of H to the full quotient Hom(Γ 22 , G)/G) entirely consists of points with associated group Z/2Z. They are "double" points in a suitable sense, which reminds "mirror" points in the language of orbifolds.

Reflections and cusp groups in HP geometry
In this section we introduce half-pipe geometry, discuss its relations with Minkowski geometry, and prove the half-pipe version of the flexibility and rigidity statements for right-angled cusp groups. 5.1. Half-pipe geometry. Let us denote by q 0 the following degenerate bilinear form on R n+1 : Then half-pipe space of dimension n is defined as Explicitly, an element A ∈ G HP n has the form Despite an inequality is involved in the definition of G HP n , this group is naturally an algebraic Lie group via the isomorphism between G HP n and O(1, n − 1) R n of Lemma 5.1, and the fact that the latter group can be defined as an algebraic subgroup of Aff(R n ) ⊂ GL(n + 1, R).
The boundary at infinity of HP n is and can be visualised as the union of a cylinder constituted by those [x] ∈ ∂HP n such that (x 0 , . . . , x n−1 ) does not vanish, and the point at infinity [e n ] ∈ ∂HP n . The latter is a distinguished point, since it is preserved by the action of every element of G HP n on ∂HP n . As usual, we consider HP n ∪HP n as a subset of RP n , the ideal closure of a subset A ⊂ HP n that is closed in HP n is its closure A in RP n , and we have HP n = HP n ∪ ∂HP n .
There is a natural map from HP n to {x ∈ HP n : x n = 0}, which is a copy of H n−1 , given simply by (x 0 , . . . , x n ) → (x 0 , . . . , x n−1 , 0). We shall call this map the projection π : HP n → H n−1 .
The map π is equivariant with respect to the obvious epimorphism G HP n → Isom(H n−1 ), and extends to a map π : HP n {[e n ]} → H n−1 . The fibres of π are called degenerate lines, since they extend to projective lines in RP n by adding the point [e n ] at infinity, and the restriction of the bilinear form b 0 associated to q 0 is degenerate. Degenerate lines are preserved by the action of G HP n . 5.2. Duality with Minkowski space. We will find comfortable to exploit the well-known duality between half-pipe and Minkowski geometry. We will not provide details of the proofs here, see [BF20, FS19, RS] for a more complete treatment.
The fundamental observation is that HP n is identified to the space of spacelike affine hyperplanes in Minkowski space R 1,n−1 := (R n , q 1 ) where q 1 is the non-degenerate bilinear form on R n introduced in Section 2.1. The correspondence is given by associating to a point x ∈ HP n the affine hyperplane of R 1,n defined by the equation b 1 ((x 0 , . . . , x n−1 ), (y 0 , . . . , y n−1 )) + x n = 0 , for b 1 the bilinear form associated to q 1 . The isometry group Isom(R 1,n−1 ) ∼ = O(q 1 ) R n acts naturally on the space of spacelike affine hyperplanes, and the correspondence is also well-behaved with respect to the group actions, as we summarise in the following lemma (see for instance [BF20], [FS19] or [RS, Lemma 2.8]).
In this work, we will adopt almost entirely this "dual" point of view for half-pipe geometry. In this setting, the boundary ∂HP n has a natural identification: where the point [e n ] in ∂HP n corresponds to ∞ on the right-hand side, while ∂HP n {[e n ]} identifies to the space of lightlike affine hyperplanes using again (23). Geometrically, the decomposition in the right-hand side of (24) reflects the fact that, up to taking a subsequence, a sequence of spacelike affine hyperplanes in R 1,n−1 may either converge to a lightlike hyperplane or escape from all compact subsets. The projection π is interpreted in this dual setting as the map which associates to a spacelike affine hyperplane in R 1,n−1 its unique parallel linear hyperplane. Equivalently, thinking of π with values in H n−1 , it associates to a spacelike affine hyperplane its normal direction with respect to the Minkowski product b 1 . Of course π extends to the complement of ∞ in ∂HP n , with values in ∂H n−1 .

Hyperplanes.
Let us now consider hyperplanes in half-pipe geometry.
Definition 5.2 (HP hyperplane). A half-pipe hyperplane is the intersection of HP n with a linear hyperplane in R n+1 . It is called degenerate if it contains a degenerate line of HP n ; non-degenerate otherwise.
From now on, we will always think of HP n dually as the space of spacelike affine hyperplanes in R 1,n−1 , using Lemma 5.1. For more details on the proofs of the following statements, see [RS,Section 4.3].
Lemma 5.3. Any non-degenerate hyperplane of HP n is dual to the set of spacelike affine hyperplanes going through a given point p ∈ R 1,n−1 . We will refer to the point p as the dual point to the non-degenerate hyperplane, and conversely we will make reference to the hyperplane dual to a point of R 1,n−1 . With this duality approach, it is very easy to describe the relative position of non-degenerate hyperplanes: Lemma 5.4. Given two points p, q ∈ R 1,n−1 , their dual hyperplanes In half-pipe geometry, the situation for degenerate and non-degenerate hyperplanes is qualitatively different, as we shall see also in Section 5.4 below. Let us first characterise degenerate hyperplanes in terms of Minkowski geometry: Lemma 5.5. Any degenerate hyperplane of HP n is the preimage of a hyperplane in H n−1 by the projection map π : HP n → H n−1 . That is, it is dual to the set of spacelike affine hyperplanes having normal direction in a given hyperplane of H n−1 .
There are three possibilities for the relative position of two degenerate hyperplanes H 1 = π −1 (S 1 ) and H 2 = π −1 (S 2 ) in HP n : • If S 1 and S 2 intersect in H n−1 , then H 1 and H 2 intersect in HP n in the subset π −1 (S 1 ∩ S 2 ); • If S 1 and S 2 intersect in ∂H n−1 , then H 1 and H 2 intersect in a degenerate line of ∂HP n ; • If S 1 and S 2 are disjoint in H n−1 , then H 1 and H 2 only intersect in ∞ ∈ ∂HP n .
Asymptoticity of two hyperplanes and asymptoticity of a hyperplane to a point at infinity are defined similarly to the hyperbolic and AdS settings (see Section 2.3).

5.4.
Reflections. Like in pseudo-Riemannian geometry, a reflection in HP n is a non-trivial involution in G HP n that fixes pointwise a hyperplane.
We shall again distinguish two cases: Proposition 5.6. There exists a unique reflection in G HP n fixing a given non-degenerate hyperplane in HP n .
Proof. By Lemma 5.3, a reflection in G HP n is induced by an element of Isom(R 1,n−1 ) that fixes setwise all the spacelike hyperplanes going through a point p ∈ R 1,n−1 . The involution φ(−id, 2p) therefore has such a property. It is the only reflection fixing the hyperplane dual to p. Indeed, for a transformation φ(A, v) with this property, the linear part A must fix all the timelike directions in R 1,n−1 , hence A = ±id, but the choice A = id implies necessarily v = 0 because φ(A, v) has order two, and therefore gives a trivial transformation.
Let us now consider degenerate hyperplanes: Proposition 5.7. There exists a one-parameter family of reflections in G HP n fixing a given degenerate hyperplane in HP n .
Proof. From Lemma 5.5, a degenerate hyperplane in HP n has the form π −1 (H X ) where, using the notation of Section 2.2, X denotes a vector in R 1,n−1 such that q 1 (X) = 1 and H X is the hyperplane in H n−1 induced by the orthogonal complement X ⊥1 . Any reflection in G HP n fixing π −1 (H X ) pointwise must be of the form φ(A, v) where the linear part A fixes X ⊥1 pointwise. Hence the only possible candidates for A are the identity and the Minkowski reflection in H X , which we denote by r X . Since (A, v) is assumed to be an involution, A = id only gives the trivial transformation (i.e. v = 0). On the other hand, imposing the involutive condition for the choice A = r X we obtain the reflections φ(r X , v) for any v ∈ Span(X). These are indeed reflections in the half-pipe hyperplane π −1 (H X ), since they fix setwise all spacelike hyperplanes of R 1,n−1 with normal direction in H X .
Finally, it is necessary to analyse conditions which assure that two reflections commute. From Proposition 5.6, it is clear that two reflections φ(−id, 2p) and φ(−id, 2q) in nondegenerate hyperplanes do not commute unless p = q, i.e. unless the hyperplanes of reflection coincide.
By Proposition 5.7, reflections in degenerate hyperplanes are induced by Minkowski reflections in timelike hyperplanes. Hence two reflections φ(r X1 , v 1 ) and φ(r X2 , v 2 ) commute if and only if their linear parts commute.
The remaining case is considered in the following lemma, which is straightforward: Lemma 5.8. Let v, w, X be vectors in R 1,n−1 , with q 1 (X) = 1 and v ∈ Span(X). The Minkowski isometries (r X , v) and (−id, w) commute if and only if w = v + u with u ∈ X ⊥1 .
Proof. An easy computation shows that (r X , v) and (−id, w) commute if and only if Writing w = λX + u for λ ∈ R and u ∈ X ⊥1 , we have r X (w) = −λX + u, hence the condition (25) is equivalent to λX = v.

5.5.
Right-angled cusp groups. Let us now discuss the properties of flexibility and rigidity of cusp representations for half-pipe geometry, similarly to what we did for hyperbolic and AdS geometry in Section 3. The statements will be completely analogous, but the proofs simpler than their AdS (and hyperbolic) counterparts above. The definitions of cusp groups and collapsed cusp groups are parallel to the AdS case: Definition 5.9 (Cusp groups for HP 3 ). The image of a representation of Γ rect into G HP 3 is called: • a cusp group if the four generators are sent to reflections in four distinct planes asymptotic to a common point in ∂HP 3 ; • a collapsed cusp group if the four generators are sent to reflections along three distinct planes, two degenerate and one non-degenerate, asymptotic to a common point in ∂HP 3 .
It follows from the discussion of the previous section that a cusp group representation must necessarily map two generators corresponding to opposite sides of the rectangle to reflections in degenerate hyperplanes, and the other two generators to reflections in non-degenerate hyperplanes.
The following example describes the structure of a (possibly collapsed) cusp group in HP geometry. By the non-uniqueness of half-pipe reflections in a degenerate plane (Proposition 5.7), we need to describe not only the planes fixed by the reflections associated to each generators, but also the reflections themselves.
Example 5.10. Let the image of ρ : Γ rect → G HP 3 be a cusp group or collapsed cusp group, let s 1 , s 2 be the generators such that ρ(s 1 ), ρ(s 2 ) are reflections in a non-degenerate plane, and t 1 , t 2 those such that ρ(t 1 ), ρ(t 2 ) are reflections in a degenerate plane. Up to conjugacy, we can assume that ρ(s 1 ) = φ(−id, 0), that is, ρ(s 1 ) is the unique reflection in the dual plane to the origin of R 1,2 .
Using Lemma 5.8, ρ(t 1 ) and ρ(t 2 ) are necessarily of the form φ(r Xi , 0), for X i a unit spacelike vector in R 1,2 . This means that the two degenerate planes fixed by ρ(t i ) are of the form π −1 (H Xi ), for i = 1, 2. Since the ideal closures of the four planes are assumed to meet in a single point in ∂HP 3 , necessarily the ideal closure of the geodesics H X1 and H X2 of H 2 meet in ∂H 2 . This means that X ⊥1 1 ∩ X ⊥1 2 is a lightlike line in R 1,2 . Finally, by Lemma 5.8 ρ(s 2 ) must be of the form φ(−id, w) for some w ∈ X ⊥1 1 ∩ X ⊥1 2 . This means that the non-degenerate plane fixed by ρ(s 2 ) is the dual of the point w/2 ∈ R 1,2 . If w = 0, then we have a collapsed cusp group, otherwise a cusp group. See Figure 7 on the left.
Let us now prove the HP analogue of Propositions 3.8 and 3.10 (see Figure 7).
Proposition 5.11. Let ρ : Γ rect → G HP 3 be a representation whose image is a cusp group or a collapsed cusp group. For all nearby representations ρ , exactly one of the following possibilities holds: (1) If s 1 and s 2 are generators such that ρ(s 1 ) = ρ(s 2 ), then ρ (s 1 ) = ρ (s 2 ).
(2) The image of ρ is a cusp group.
(3) A pair of opposite planes intersect in HP 3 , while the other pair of opposite planes have disjoint ideal closures in HP 3 .
Proof. Let ρ : Γ rect → G HP 3 be a representation nearby ρ. As in Example 5.10, we can assume that the reflection associated to one of the generators s 1 of Γ rect is ρ (s 1 ) = φ(−id, 0), so that its fixed plane is the dual plane to the origin of R 1,2 . Repeating the argument of Example 5.10, we have ρ (t i ) = φ(r Xi , 0) for some unit spacelike vectors X i , and ρ (s 2 ) = φ(−id, w) for some w ∈ X ⊥1 1 ∩ X ⊥1 2 . If w = 0, we are in case (1). Let us therefore assume w = 0. If the ideal closures of the geodesics H X1 and H X2 intersect in ∂H 2 , then X ⊥1 1 ∩ X ⊥1 2 is a lightlike geodesic, hence the image of ρ is a cusp group as in Example 5.10 and we are in case (2).
If H X1 and H X2 intersect in H 2 , then X ⊥1 1 ∩ X ⊥1 2 is a timelike geodesic, hence w is timelike. By Lemma 5.4, the fixed planes of ρ (s 1 ) and ρ (s 2 ) are disjoint, while the degenerate hyperplanes fixed by ρ (t 1 ) and ρ (t 2 ), namely π −1 (H X1 ) and π −1 (H X2 ), intersect in HP 3 (along a degenerate line). Hence point (3) is fulfilled. Proposition 5.11. In red, two non-degenerate planes, in blue two degenerate planes, and in green their intersections, which are geodesics in a copy of H 2 . On the left, the green geodesics are asymptotic and we have a cusp group. In the middle, they are ultraparallel, so the degenerate blue planes of HP 3 are disjoint, while the non-degenerate red planes intersect. On the right, the green geodesics intersect, so do the blue (degenerate) planes, while the red (non-degenerate) planes are disjoint.
Moving to dimension four, we define cusp groups in half-pipe geometry: Definition 5.12 (Cusp groups for HP 4 ). The image of a representation of Γ cube into G HP 4 is called: • a cusp group if the 6 generators are sent to reflections in 6 distinct hyperplanes asymptotic to a common point at infinity; • a collapsed cusp group if the 6 generators are sent to reflections along five distinct hyperplanes, four degenerate and one spacelike, asymptotic to a common point at infinity.
The half-pipe version of Proposition 3.17 and 3.19 is now proved along the same lines: Proposition 5.13. Let ρ : Γ cube → G HP 4 be a representation whose image is a cusp group or a collapsed cusp group. For all nearby representations ρ , exactly one of the following possibilities holds: (1) If s 1 and s 2 are generators such that ρ(s 1 ) = ρ(s 2 ) is a reflection in a non-degenerate hyperplane, then ρ (s 1 ) = ρ (s 2 ).
(2) The image of ρ is a cusp group.
Proof. Let us denote by s 1 and s 2 the generators of Γ cube (corresponding to opposite faces of the cube) that are sent by ρ to reflections in a non-degenerate hyperplane; by t 1 , t 2 and u 1 , u 2 the other two pairs of opposite generators, which are necessarily sent to reflections in degenerate hyperplanes. By continuity, the same holds for ρ .
Up to conjugation we can assume that ρ (s 1 ) = φ(−id, 0), and therefore by Lemma 5.8 ρ (t i ) = φ(r Xi , 0) and ρ (u i ) = φ(r Yi , 0), for X i , Y i unit spacelike vectors. The restriction of ρ to the subgroup generated by these four elements gives a representation of Γ rect in a copy of Isom(H 3 ), and is nearby a 3-dimensional cusp group.
Suppose first ρ | Γrect is a cusp group in Isom(H 3 ). This means that X ⊥1 is a lightlike line in R 1,3 . Then ρ (s 2 ) is of the form φ(−id, w) and by Lemma 5 Hence ρ gives a cusp group in G HP 4 and we are in point (2). If ρ | Γrect does not give a cusp group in H 3 , by Proposition 3.8 two planes intersect in H 3 , while the ideal closures of the other two are disjoint in H 3 . We will derive a contradiction. Up to relabelling, we can assume that the planes H X1 and H X2 intersect in H 3 , while the closures of H Y1 and H Y2 are disjoint. Hence in the degenerate subspace π −1 (H X1 ) (which is a copy of HP 3 ), the sets π −1 (H Y1 ) ∩ π −1 (H X1 ) and π −1 (H Y2 ) ∩ π −1 (H X1 ) are disjoint. Applying Proposition 5.11 to the restriction of ρ to the subgroup generated by s 1 , s 2 , u 1 , u 2 , the fixed planes of ρ (s 1 ) and ρ (s 2 ) intersect in π −1 (H X1 ) (and thus in HP 4 ).

Group cohomology and the HP character variety of Γ 22
The goal of this section is to prove the half-pipe part of Theorem 4.16. An essential step is an explicit computation of the first cohomology group H 1 0 (Γ 22 , R 1,3 ) in Proposition 6.5, a result for which we will give other applications in Section 7.
6.1. Preliminaries on group cohomology. We recall here a few notions of group cohomology.
Let Γ be a group, V a finite-dimensional real vector space, and : Γ → GL(V ) a representation. The first cohomology group of Γ associated to is the quotient • the space of coboundaries is When Γ is a right-angled Coxeter group, the space Z 1 (Γ, V ) has the following description in terms of generators and relations.
Lemma 6.2. Let Γ be a right-angled Coxeter group as in Defintion 3.1, and : Γ → GL(V ) a representation. Then Z 1 (Γ, V ) is isomorphic to the vector space of functions τ : S → V such that: • τ (s) ∈ Ker(id + (s)) for all s ∈ S, and Proof. Clearly a cocycle in Z 1 (Γ, V ) is determined by its values on the generators. The conditions that have to be satisfied by τ for each relation are 0 = τ (s 2 ) = (s)τ (s) + τ (s), from which we get the first point, and τ (s i s j ) = τ (s j s i ) for every (s i , s j ) ∈ R. Expanding τ (s i s j ) = (s i )τ (s j ) + τ (s i ) we obtain the second point.
6.2. A curve of geometric representations. We now introduce the half-pipe representations of our interest, which have been computed in [RS,Remark 7.16] by applying a rescaling argument to the hyperbolic or AdS holonomy representations ρ t .
Notation. Throughout the following, we will denote by ·, · the Minkowski product of R 1,3 (previously denoted by b 1 ) and by v ⊥ ⊂ R 1,3 the orthogonal complement of v ∈ R 1,3 with respect to the Minkowski product.
Definition 6.3 (The HP representation ρ λ ). Given λ ∈ R, we define a representation on the standard generators of Γ 22 as follows. The linear part 0 is independent of λ and is defined by: while the translation part is: where the vectors v s are defined in Table 4.
Recall that the vectors in Table 4 definine the bounding planes of an ideal right-angled cuboctahedron in H 3 . Moreover, r v denotes the reflection in O(1, 3) in the hyperplane v ⊥ , namely, the linear transformation acting on v ⊥ as the identity and on the subspace generated by v as minus the identity.
Remark 6.4. When λ = 0, the representation ρ 0 is naturally identified to those introduced in Definition 4.1 for t = 0. Indeed, recall from Section 4.3 that in the hyperbolic and AdS case ρ 0 takes value in the stabiliser G 0 of the hyperplane {x 4 = 0}, and G 0 is a common subgroup of Isom(H 4 ) and Isom(AdS 4 ), both seen as a subgroups of GL(5, R). Now, the representation ρ 0 = ( 0 , 0) introduced in Definition 6.3 also takes value in the stabiliser of {x 4 = 0} in G HP 4 which coincides again with the subgroup G 0 of GL(5, R). Under the isomorphism with Isom (R 1,3 ), the group G 0 is dually identified with the stabiliser of the origin in R 1,3 , namely the linear subgroup O(1, 3) < Isom (R 1,3 ). The explicit isomorphism O(1, 3) ∼ = G 0 is given by where the sign ± is positive if A preserves H 3 ⊂ R 1,3 , and negative otherwise. Under the isomorphism (28), the representation ρ 0 = ( 0 , 0) of Definition 6.3 (with zero translation part) coincides with the "collapsed" representation expressed in (6). This justifies that in the statement of Theorem 1.1 we refer to the same representation ρ 0 in all three geometries.
The goal of the following section is to compute the first cohomology group associated to the representation 0 : Γ 22 → O(1, 3) of Definition 6.3. Applications of the result will then be given in Sections 6.4 and 7.2.
6.3. The geometric cocycle is a generator. Recall Definition 6.3. The goal of this section is to prove the following: Proposition 6.5. The vector space H 1 0 (Γ 22 , R 1,3 ) has dimension one. To prove Proposition 6.5, we will show that every cohomology class in H 1 0 (Γ 22 , R 1,3 ) is represented by a cocycle τ λ of the form (27), for some λ ∈ R.
Let us observe that τ λ vanishes on all the letter generators and that τ λ i − and τ λ i + are all vectors of norm |λ| for the Minkowski product on R 1,3 , since all the v i have unit Minkowski norm.
Proof. By Lemma 6.2 we get τ (i − ) ∈ Ker(id + 0 (i − )). This kernel equals the subspace generated by v i since 0 (i − ) is the Minkowski reflection fixing the hyperplane v ⊥ i . The proof for the letter generators is the same.
The following step is a first reduction of the problem.
Lemma 6.8. Let τ ∈ Z 1 0 (Γ 22 , R 1,3 ). Then there exists a unique η ∈ B 1 0 (Γ 22 , R 1,3 ) such that, After the proof of Lemma 6.8, we will show that ifτ satisfies (29), then it is of the form (27) for some λ ∈ R. Together with Lemma 6.8, this will imply that and therefore that H 1 0 (Γ 22 , R 1,3 ) is one-dimensional. Proof of Lemma 6.8. Let τ be any cocycle. By Lemma 6.7, we have that τ (X) ∈ Span(v X ) for all X ∈ {A, B, C, D}. Define the linear map The proof follows if we show that L is invertible. Let us write the matrix associated to L in the basis {v A , v B , v C , v D } on the source and on the target. Recalling that the v X are all unit vectors for the Minkowski product ·, · and that ρ 0 (X) is the reflection in v ⊥ X , we have which is invertible by the non-degeneracy of the Minkowski product.
Proof. Let us show the first point, the second being completely analogous. By Lemma 6.2 where we have used that 0 (i + ) = −id, that 0 (j − ) is the reflection in the Minkowski hyperplane v ⊥ j , and that τ (j − ) ∈ Span(v j ) by Lemma 6.7. Hence τ (i + ) − τ (j − ) is in the kernel of id − 0 (j − ), namely in v ⊥ j .
Proof. It follows from Lemma 6.7 thatτ (0 − ) = µ 0 v 0 , and similarlyτ (3 − ) = µ 3 v 3 . We remark that we have no similar condition on the i + coming from the relation that i + squares to the identity.
Remark 6.12. The proof of Lemma 6.11 only worked for i = 0, 3 because we used thatτ vanishes on A, B, C and D, and we needed to pick three linearly independent vectors among these four. Once we show thatτ also vanishes on E and F (Lemma 6.13 below), the same argument will apply exactly in the same way to show that for i even. This will therefore conclude the proof thatτ is in the form (27).
Observe that v A , v C and v E are linearly independent, and they are all orthogonal to v 1 . Hence {v 1 , v A , v C , v E } is a (non-orthogonal!) basis of unit vectors and we can decompose: (We ultimately will get, at the end of the proof, that λ 1 = −λ and α = γ = = 0, but we do not know this yet.) As a preliminary remark, observe thatτ (1 − ) = λ 1 v 1 , since from the relation 1 + 1 − = 1 − 1 + we obtainτ 1 , and comparing with the above decomposition, necessarilyτ (1 − ) = λ 1 v 1 .
The proof that f = 0 follows the same lines, applied to 4 + in place of 1 + , with the letters B, D and F , and in the final part to 5 − in place of 2 − .
Having shown thatτ (X) = 0 for every X, it remains to show thatτ (i + ) =τ (i − ) has the form of (27). For i = 0, 3, this is the content of Lemma 6.11. Following the same proof, one shows first thatτ for every i (it suffices to modify the proof by picking three letters X, Y and Z so that v X , v Y and v Z are orthogonal to v i ). Then using the crossed relations i + j − = j − i + -it is easy to see that there are indeed enough of such relations -one mimics the second part of Lemma 6.11 and obtains thatτ This concludes the proof of Proposition 6.5, namely that dim H 1 0 (Γ 22 , R 1,3 ) = 1. 6.4. Topology of the neighbourhood U. We are ready to conclude the proof of our weak version of Theorem 1.1, namely Theorem 4.16, in the half-pipe case. The proof of Theorem 1.1 will be completed in Section 7.
As a preliminary setup, recall that G HP 4 ∼ = O(1, 3) R 1,3 , one has a natural map: which associates to the conjugacy class of a representation ρ : Γ → G HP 4 the conjugacy class of the linear part of ρ. Recalling Lemma 6.1, one has the identification (1, 3), then H 1 (Γ, R 1,3 ) and H 1 (Γ, R 1,3 ) are isomorphic by means of the map τ → h • τ .
Proof of Theorem 4.16 -HP case. The proof follows a similar strategy to the AdS (and hyperbolic) case, so we will split again the proof in several steps which are parallel to those given in Section 4.8. Most steps are much simpler here.
Step 1 : Let us define the vertical component V in X(Γ 22 , G HP 4 ) as L −1 ([ 0 ]), namely, V consists of all the conjugacy classes of representations with linear part in [ 0 ]. By (35), V is identified to H 1 0 (Γ 22 , R 1,3 ), hence is homeomorphic to a line by Proposition 6.5. By construction, V contains the holonomy of the half-pipe orbifold structures we built in [RS].
Again, it is straightforward to check that the induced map is well-defined and injective. The representation ρ 0 is clearly in the image of Ψ, since ρ 0 = Ψ η0 where η 0 is the holonomy representation of the complete hyperbolic orbifold structure of the cuboctahedron. As in the AdS case, [η 0 ] has a neighborhood H 0 inX(Γ co , Isom(H 3 )) homeomorphic to R 12 and on which Ψ is a homeomorphism onto its image, and we define H to be the image of H 0 .
Step 3 : Clearly, the intersection of H and V consists only of the point [ρ 0 ], since any element in H has trivial translation part (up to conjugacy).
Step 4 : We now show that the point [ρ 0 ] ∈ X(Γ 22 , G HP 4 ) has a neighbourhood U which is contained in the union of the two components V and H.
We claim that if two distinct generators which are sent by ρ 0 to −id (hence necessarily of the form i + and j + ) are sent by ρ to the same reflection, than all the generators 0 + , . . . , 7 + are sent by ρ to the same reflection. In other words, if τ (i + ) = τ (j + ) for some i + = j + , then τ (i + ) = τ (j + ) for all i + , j + .
Assuming the claim, the proof then follows by the following argument. We can assume (up to conjugation) that τ (i + ) = 0 for all i + ∈ {0 + , . . . , 7 + }. By Proposition 5.13, if some of the collapsed cusp groups of ρ 0 is not deformed to a cusp group, then up to conjugation ρ has the property that ρ(i + ) = (−id, 0) for all i + ∈ {0 + , . . . , 7 + }, and therefore [ρ] ∈ H. On the other hand, if all the collapsed cusp groups of ρ 0 are deformed in ρ to cusp groups, then the linear part of ρ is of the form Lρ = Ψ η for a representation η : Γ co → Isom(H 3 ) which sends all peripheral groups to (three-dimensional) cusp groups in H 3 , and therefore η is conjugate to η 0 in Isom(H 3 ) by the the Mostow-Prasad rigidity. Thus [Lρ] = [Lρ 0 ], which means that [ρ] ∈ V.
Similarly to the AdS case, one argues similarly for 3 + and then for all the other generators, to show that ρ(i + ) = (−id, 0) for each generator i + ∈ {0 + , . . . , 7 + }, and this concludes the claim.
Step 5 : In summary, we showed that [ρ 0 ] has a neighborhood U in X(Γ 22 , Isom(AdS 4 )) which only consists of points of H and V. Additionally, one can repeat the same reasoning in the first part of the previous step, to show that for any other [ρ 0 ] in V (hence having the same linear part as ρ 0 and non-vanishing translation part) a neighbourhood of [ρ 0 ] is contained in V, as a consequence of the half-pipe cusp rigidity of Proposition 5.13 (the non-collapsed case). Hence by taking the union of all these neighbourhoods, one finds a U containing [ρ 0 ] such that U = V ∪ H.
Step 6 : For the last statement, it is evident that conjugation by Z/2Z ∼ = G HP 4 /G + HP 4 acts by switching sign to the x 13 -coordinate, since conjugation by (−id, 0), whose class generates Z/2Z, acts on H 1 0 (Γ 22 , R 1,3 ) by changing the sign. This concludes the proof.

Smoothness and transversality
In this section we complete the proof of Theorem 1.1, showing the smoothness and transversality of the two components V and H of the neighbourhood U of [ρ 0 ] in X(Γ 22 , G).
To that purpose, we first need to study the cohomology group H 1 Ad ρ0 (Γ 22 , g), complementing and using the results of Section 6.3. Then, we conclude the proof by an application of the implicit function theorem. 7.1. Preliminaries. Let G be Isom(H 4 ), Isom(AdS 4 ) or G HP 4 , and g be its Lie algebra. We shall apply the definition of first cohomology group given in Section 6.1 to the representation Ad ρ 0 : Γ 22 → GL(g) , which is the composition of our ρ 0 : Γ 22 → G and the adjoint representation Ad : G → GL(g).
In general, for a finitely presented group Γ with a given presentation with s generators and r relations, the set Hom(Γ, G) is identified to a subset of G s defined by the vanishing of r conditions given by the relations. If we encode these conditions by F : G s → G r , so as to identify Hom(Γ, G) with F −1 (0), then it is known from [Wei64] that Z 1 Ad ρ (Γ, g) is naturally identified with the kernel of dF at ρ. The isomorphism between these two vector spaces essentially associates to a germ of paths at ρ represented by t → ρ t the cocycle τ defined by which is therefore interpreted as an infinitesimal deformation of ρ.
Remark 7.1. In general, the Zariski tangent space at ρ of the real variety associated to Hom(Γ, G) is only a subspace of Z 1 Ad ρ (Γ, g). We will show that in our situation for Γ 22 they coincide at the point ρ 0 .
Let us now look at the coboundaries. It was observed in [Wei64] (see also [JM87, Lemma 2.2]) that the subspace of Ker(dF ) corresponding to the tangent space to the G + -orbit of ρ identifies to B 1 Ad ρ (Γ, g) under the correspondence (36). Indeed, by a straightforward computation, the differential of the orbit map G → Hom(Γ, G) defined by g → gρg −1 maps an element X ∈ g to the coboundary τ (γ) = X − Adρ(γ)X. Observe that in our setting, by Lemma 4.2, the action of G is not free at ρ; but the action of the identity component of G, namely G + , is indeed free. As we will see below (Lemma 7.4), this implies by a standard argument that B 1 Ad ρ0 (Γ 22 , g) ∼ = g. We conclude by stating a smoothness criterion used by Weil [Wei64, Lemma 1], essentially consisting of an application of the implicit function theorem. We refer to [KS10, Section 2.2] for more details.
Let C be an algebraic subset of Hom(Γ, G) containing ρ, say obtained by adding k extra polynomial equations. We identify C withF −1 (0), for someF : G s → G r+k compatible with F . Suppose that a neighbourhood of ρ in C is a smooth submanifold of G s of the same dimension of the kernel K of dF at ρ. Then the following hold: at the point ρ, the real variety associated to C is smooth, its Zariski tangent space is isomorphic to K (and not to a proper subspace, compare with Remark 7.1) and is naturally identified with the tangent space of C as a submanifold.
Recall that ρ 0 coincides with the composition of the representation 0 : Γ 22 → G 0 defined in (26) with the inclusion G 0 → G.
Let us look at the first factor of the decomposition (41) of H 1 Ad ρ0 (Γ 22 , g).
Proof. We claim that the group H 1 Ad 0 (Γ 22 , isom(H 3 )) is isomorphic to H 1 Ad ι (Γ co , isom(H 3 )), where Γ co is the reflection group of the right-angled cuboctahedron and ι is its inclusion into Isom(H 3 ). The latter has dimension 12, since the character variety of Γ co in Isom(H 3 ) is smooth and 12-dimensional near [ι]. We have already mentioned (in Section 4.8, Step 2) that this last fact is true by "reflective hyperbolic Dehn filling".
The second factor of the decomposition (41) of H 1 Ad ρ0 (Γ 22 , g) has already been computed in Proposition 6.5. We have in particular: Corollary 7.3. The vector space H 1 Ad ρ0 (Γ 22 , g) has dimension 13. We deduce the dimensions of the spaces of cocycles and coboundaries from the following simple lemma, which will also be used in the next section.
Proof. The orbit map is injective as a consequence that the G + -action is free (Lemma 4.2). To see that its differential at any point is injective, suppose by contradiction that a non-zero vector is in the kernel of the differential. Acting by left multiplication on G + , one then finds a nonvanishing vector field on G + which is, at any point, in the kernel of the differential. Hence the orbit map would be constant on any integral path of this vector field, thus contradicting injectivity.
Proof. By Lemma 7.4, B 1 Ad ρ0 (Γ 22 , g), which is the tangent space to the orbit, is isomorphic to g and thus has dimension 10. Combining this with Corollary 7.3, we conclude that Z 1 Ad ρ0 (Γ 22 , g) has dimension 10. Remark 7.6. One could also check directly that dim B 1 Ad ρ0 (Γ 22 , g) = 10: from Remark 6.9 the second factor in the decomposition (40) has dimension 4, and by a similar argument one can prove that the first factor has dimension 6. 7.3. Conclusion of the proofs. We can now prove the main result of the section.
Recall that G is Isom(H 4 ), Isom(AdS 4 ) or G HP 4 , and g is its Lie algebra. We denote as usual by U, V, H ⊂ Hom(Γ 22 , G) the preimages of U, V, H ⊂ X(Γ 22 , G), respectively. All these sets are defined and studied in the proof of Theorem 4.16 in Sections 4.8 and 6.4. Recall that U = V ∪ H and V ∩ H = {[ρ 0 ]}.
For brevity, in the following, given a real affine algebraic set S, by "Zariski tangent space to" (resp. "component of") S we refer to the Zariski tangent space to (resp. a component of) the real variety associated to S.
Theorem 7.7. The sets V and H are smooth components of U, of dimension 11 and 22, respectively. Moreover, V ∩ H is the G-orbit of ρ 0 , and the Zariski tangent spaces of V and H at ρ 0 intersect transversely in the Zariski tangent space of Hom(Γ 22 , G) at ρ 0 .
Proof. We first show that V is a smooth component of U by applying the smoothness criterion stated at the end of Section 7.1. Note that V is the intersection of U with the algebraic subset Hom 0 (Γ 22 , G) of Hom(Γ 22 , G) (see Definition 4.8 and the discussion below). In particular V is a neighbourhood of ρ 0 in Hom 0 (Γ 22 , G).
Moreover, V is a smooth 11-dimensional manifold. This is true in the hyperbolic or AdS case by Proposition 4.11, and in the HP case since there V is the total space of a smooth vector bundle with fibre Z 1 0 (Γ 22 , R 1,3 ) and base the Isom(H 3 )-orbit of 0 (hence a rank-5 bundle over a 6-manifold). Recall indeed that Z 1 0 (Γ 22 , R 1,3 ) has dimension 5 by Proposition 6.5 and Remark 6.9.
Let us now look at H. Note that it is the intersection of U with an algebraic subset of Hom(Γ 22 , G). Combining our study of H in Sections 4.8 and 6.4 with Lemma 7.4, we get that H is a smooth 22-dimensional manifold. Moreover, T ρ0 H is contained in Z 1 Ad 0 (Γ 22 , isom(H 3 ))⊕ B 1 0 (Γ 22 , R 1,3 ) under (36) and (39). The proof of Proposition 7.2 shows that the first factor is isomorphic to Z 1 Ad ι (Γ co , isom(H 3 )). The latter is the space of infinitesimal deformations of the right-angled cuboctahedron, and