Documenta Math. 373 Asymptotic Behavior of Word Metrics on Coxeter Groups

We study the geometry of tessellation defined by the walls in the Moussong complex M W of a Coxeter group W. It is proved that geodesics in M W can be approximated by geodesic galleries of the tessellation. A formula for the translation length of an element of W is given. We prove that the restriction of the word metric on the W to any free abelian subgroup A is Hausdorff equivalent to a regular norm on A.


Introduction
For any Coxeter system (W, S), Moussong constructed a certain piecewise Euclidean complex M W on which W acts properly and cocompactly by isometries [Mou88].This complex is complete, contractible, has nonpositive curvature and the Cayley graph C W of W (with respect to S) is isomorphic to the 1-skeleton of M W .A wall in M is the fixed-point set of a reflection in W. It turns out that the walls are totally geodesic subspaces in M W and each wall divides M W into two path components.The set of all walls defines a wall tessellation of M. The set of all tiles (=chambers) of this tessellation together with an appropriate adjacency relation is isomorphic to the Cayley graph C W .We shall prove that geodesics in M W can be uniformly approximated by geodesic galleries of the wall tessellation (= geodesic paths in C W ) (Theorem 3.3.2).This approximation result allows us to prove that for any "generic" element w ∈ W of infinite order there is a conjugate v which is straight i.e., ℓ(v n ) = nℓ(v) for all n ∈ N, Documenta Mathematica 16 (2011)   where ℓ(v) is the word length on W (Theorem 4.1.5).There is a constant c = c(W ), such that for any w ∈ W of infinite order there is a conjugate v of w c , which is straight (Theorem 4.1.6).The restriction of the word metric on W to any free abelian subgroup A is Hausdorff equivalent to a regular norm on A (Theorem 4.3.2).Acknowledgment.I would like to thank Mathematisches Institut der Heinrich-Heine-Universitat for the hospitality during 1999.Thanks to H. Abels, B. Brink, M. Davis, M. Gromov, F. Grunewald, R. Howlett and E. B. Vinberg for their help.This research was supported by a DFG grant Gr 627-11 grant and by SFB 343, SFB 701 of the DFG at the University of Bielefeld.

Preliminaries on Moussong complexes
To any Coxeter system (W, S) one can canonically associate the Moussong complex M = M W , which is a piecewise Euclidean complex with W as the set of vertices.Their cells are Euclidean polyhedra, which are the convex hulls of sets, naturally bijected with the spherical cosets of W .In particular, the 1-cells of M are in bijective correspondence with the sets {w, ws}, where w ∈ W and s ∈ S. Hence the 1-skeleton of M is nothing but a modified Cayley graph of W with respect to S ( the modification consists in identifying an edge w s → ws with its inverse ws s → w).W acts cellularly and isometrically on M W and this induces the standard W -action on the Cayley graph of W .In the next subsections we carry out in detail the construction of M W following the thesis of D. Krammer [Kra94].

Coxeter groups
A Coxeter system is a pair (W, S) where W is a group and where S is a finite set of involutions in W such that W has the following presentation: s : s ∈ S|(ss ′ ) m ss ′ = 1 when m ss ′ < ∞ , where m ss ′ ∈ {1, 2, 3, . . ., ∞} is the order of ss ′ , and m ss ′ = 1 if and only if s = s ′ .We refer to W itself as a Coxeter group when the presentation is understood.The number of elements of S is called its rank.The Coxeter system (W, S) is called spherical if W has finite order.A subgroup of W is called special if it is generated by a subset of S. For each T ⊆ S, W T denotes the special subgroup generated by T .Any conjugate of such a subgroup will be called parabolic.A remarkable feature of Coxeter systems is that for any subset T ⊆ S the pair (W T , T ) is a Coxeter system in its right and moreover a presentation of W T is defined by the numbers m tt ′ , t, t ′ ∈ T .If (W S , S) is a Coxeter system of finite rank then we write V S for the real vector space with a basis of elements (e s ) for s ∈ S. Put a symmetric bilinear form B on V S by requiring: B(e s , e s ′ ) = − cos(π/m ss ′ ).
Documenta Mathematica 16 (2011) 373-398 (This expression is interpreted to be −1 in case m(s, s ′ ) = ∞.)Evidently B(e s , e s ) = 1, while B(e s , e s ′ ) ≤ 0 if s = s ′ .Since e s is non-isotropic, the subspace H s = e ⊥ s orthogonal to e s is complementary to the line Re s .Associated to s ∈ S is an automorphism a s of B acting as the reflection v → v − 2(v, e s )e s in the hyperplane e ⊥ s .The result by Tits asserts that the correspondence s → a s extends to a faithful representation of W as a group of automorphisms of the form B. (cf.[Bou], Ch.V, s.4).

Trading Coxeter cells
The Coxeter group W is finite if and only if the form B(e s , e s ′ ) is positive definite.We call a set J ⊆ S spherical if W J is finite or, equivalently, the restriction of the form B to the subspace V J = j∈J R e j is positive definite.Let J ⊆ S be spherical.Since V J is non-degenerate, there exists a unique basis {f J j |j ∈ J} of V J dual to {e j : j ∈ J} with respect to B. A space V J that comes equipped with a positive definite inner product B|V J will be denoted by E J and called the Euclidean space associated to J. Define the Coxeter cell X J to be the convex hull of the W J -orbit: For convenience we define W ∅ = {1} and X ∅ = {0}the origin of E J .More generally, for any spherical K and any J ⊆ K we consider the faces of the polyhedron X K = Ch(W K x K ) of the form We do not exclude the case J = ∅, where X ∅K = {x K }.We call the extremal points of the cell X J the vertices.For spherical J ⊆ S, let p J : V S → E J denote the orthogonal projection.It is well defined since the quadratic form on E J is non-degenerate.
The dimension of the cell X J equals the cardinality of J.For spherical subsets J ⊆ K of S we have The nonempty faces of X K are precisely those of the form wX JK (J ⊆ K, w ∈ W K ).In particular, the vertex set of X J is precisely W J x J .
Example 1.2.2 1) If J = {j} then f J j = e j and X J = Ch(e j , −e j ) is a line segment.2) Let J = {s, s ′ } be spherical, so w = ss ′ has finite order m ss ′ .Set V s,s ′ = Re s + Re s ′ .The restriction of B to V s,s ′ is positive definite and both s and s ′ act as orthogonal reflections in the lines Rf s , Rf s ′ respectively.Since B(e s , e s ′ ) = − cos(π/m ss ′ ) = cos(π − (π/m ss ′ )), the angle between the Documenta Mathematica 16 (2011) 373-398 G. A. Noskov rays R + e s and R + e s ′ is equal to π − (π/m ss ′ ), forcing the angle between the reflecting lines Rf s , Rf s ′ to be equal π/m ss ′ .The vectors f s , f s ′ are of the same length, lie in the cone R + e s + R + e s ′ ; moreover, f s + f s ′ is a bisectrix between the reflecting lines Rf s , Rf s ′ hence the convex hull of the orbit

Gluing the Moussong complex
Now we build the Moussong complex of W = W S as follows.Take the union Introduce an equivalence relation R on U , generated by the following gluing relations: 2. The cells (w, X K ), (w, X L ) are glued along the face (w, X J ), J = K ∩ L, which is embedded into each of them (by the map p J ) as (w, X JK ) and (w, X JL ) respectively .
The quotient space of U modulo R is called the Moussong complex of W and is denoted by M W .The group W acts on U by u(w, x) = (uw, x).This action respects the relation R and hence induces a cellular action of W on M W .With some abuse in notation we will denote the natural image of (1, X J ) in M by X J , so any cell in M is of the form wX J for some w ∈ W, J ⊆ S. We call J the type of the cell wX J .There is a distinguished vertex x 0 = X ∅ in M. Note that x J = x 0 for any spherical J (by condition (2)).
It can be shown that the inclusion maps of the cells are injective, see [Kra94].
The canonical metric in each cell allows to measure the lengths of finite polygonal paths in M. The path metric d on M is defined by setting the distance between x, y ∈ M to be the infimum of the lengths of polygonal paths joining x to y.We summarize the main properties of M in the following theorem.
Theorem 1.3.1 ([Kra94], [Mou88]) Relative to the path metric M is a contractible, complete, proper CAT(0) space.The Coxeter group W acts on M cellularly and this action is isometric, proper and cocompact.This action is simply transitive on the set of vertices M (0) of M, in particular M (0) coincides with W x 0 .
For the convenience of the reader we repeat the relevant definitions.A geodesic, or geodesic segment, in a metric space (X, d) is a subset isometric to a closed interval of real numbers.Similarly, a loop S 1 → X is a closed geodesic if it is an isometric embedding.( Here S 1 denotes the standard circle equipped with its arc metric, possibly rescaled so that its length can be arbitrary).We say that X is a geodesic metric space if any two points of X can be connected by a geodesic.We denote by [x, y] any geodesic joining x and y.We will always parameterize [x, y] by t → p t (0 ≤ t ≤ 1), where d(x, p t ) = td(x, y) for all t.Given three points x, y, z in X, the triangle inequality implies that there is a comparison triangle in the Euclidean plane R 2 , whose vertices x, y, z have the same pairwise distances as x, y, z.Given a geodesic [x, y] and a point p = p t ∈ [x, y], there is a corresponding point p = p t on the line segment [x, y] in R 2 .A geodesic metric space X is called a CAT(0) space if for any x, y in X there is a geodesic [x, y] with the following property: For all p ∈ [x, y] and all z ∈ X, we have with z and p as above.Let X be a CAT(0) space.Then there is a unique geodesic segment joining each pair of points , then M W is an infinite n-regular tree with edges of length 2.
Lemma 1.3.3Any cell of a CAT (0) piecewise Euclidean complex X is isometrically embedded into X.In view of uniqueness of geodesics this is equivalent to the convexity of a cell.

G. A. Noskov
Proof.We have to show is that for any two points a, b of a cell C the Euclidean arc α in C between them is a global geodesic.We may assume that C is of minimal dimension.For any two points x and y in the interior of α the closed subarc β ⊂ α between x and y lies in the interior of C. Clearly there is an ǫ > 0, such that for any cell C ′ , having C as a face, the distance from β to the set

The action of reflections on cells
We refer to the notation of §1.3.
Lemma 1.4.1An element w ∈ W leaves the cell uX K invariant if and only if u −1 wu ∈ W K .In the latter case w acts on the X K -coordinate of ux ∈ uX K as the element u −1 wu ∈ W K .
Proof.Indeed, the cell uX K is uniquely determined by its set of vertices uW K x 0 and it is w-invariant if and only if uW K is w-invariant under left translation.The latter happens if and only if wuW The second assertion follows from the equality w(ux) = u(u −1 wux).
Lemma 1.4.2(An "overcell" of invariant cell is invariant too.)If C ⊆ C ′ are cells and wC = C for some w ∈ W, then wC ′ = C ′ .
Proof.Writing C = uX J with w ∈ W, J ⊆ S we can represent C ′ in the form C ′ = uX K , J ⊆ K.By Lemma 1.4.1 wC = C implies u −1 wu ∈ W J and thus u −1 wu ∈ W K .Again by the same lemma wC ′ = C ′ .
Definition 1.4.3Let (W ; S) be a Coxeter system.A reflection in W is an element that is conjugate in W to an element of S.
Lemma 1.4.4For any cell C of M and any reflection w ∈ W either C ∩ wC is empty or else w acts as a reflection on C.
Proof.Suppose that the cell C ∩ wC is nonempty.Then it is invariant under the action of w.Since it is a face of C, by Lemma 1.4.2we conclude that wC = C. Now by Lemma 1.4.1 w ∈ W acts as a reflection on C.

Angles and geodesics in M
The notion of angle in an arbitrary piecewise Euclidean complex can be defined in terms of the link distance, see e.g.[BB97].Namely, let X be a piecewise Euclidean complex, x ∈ X and let A be a Euclidean cell of X containing x.The link lk x A of is the set of unit tangent vectors ξ at x such that a nontrivial line segment with the initial direction ξ is contained in A. We define the link lk x X by lk x X = ∪ A∋x lk x A, where the union is taken over all closed cells containing x.
Recall that the CAT(0)-condition for X is equivalent to the following (see e.g. [BB97]): 1. X is 1-connected and 2. The length of any geodesic loop in the link of any vertex of X is greater or equal to 2π.
Here S 1 denotes the standard circle equipped with its arc metric (possibly rescaled so that its length can be arbitrary).Angles in lk x A induce a natural length metric d x on lk x S, which turns lk x S into a piecewise spherical complex.For ξ, η ∈ lk x X define ∠(ξ, η) = min(d x (ξ, η), π).Now any two segments σ 1 , σ 2 in X with the same endpoint x have the natural projection image in lk x X and we define ∠ x (σ 1 , σ 2 ) to be the angle between these two projections.We will use the following criterion of geodesicity: Lemma 1.5.1 ([BB97]) Let X be a piecewise Euclidean CAT(0)-complex.If each of the segments σ 1 , σ 2 is contained in a cell and σ 1 ∩ σ 2 = {x}, where x is an endpoint of each of the segments, then the union An m-chain from x to y is an (m + 1)-tuple T = (x 0 , x 1 , . . ., x m ) of points in X such that x = x 0 , y = x m and each consecutive pair of points is contained in a cell.Every m-chain determines a polygonal path in X, given by the concatenation of the line segments [x i , x i+1 ], i = 0, ..., i = m.An m-taut chain from x to y is an m-chain such that 1. there is no triple of consecutive points contained in a cell and 2.
(2) the union of two subsequent segments is geodesic in the union of cells, containing these segments.
(The union is equipped with its path metric).Note that if a chain is taut then only its first and last entries lie in the interior of a top dimensional simplex of X.The result of M. Bridson asserts that if X is a piecewise Euclidean complex then X with its path metric is a geodesic space and the geodesics are the paths determined by taut chains [BH99, Theorem.7.21].

G. A. Noskov 2 Walls in the Moussong complex
The notion of wall in the Moussong complex (as well as in the Coxeter complex) can be defined as the fixed-point set of reflection from the underlying Coxeter group.On the other hand they can be defined as the equivalence classes of "midplanes" (which are the fixed-point sets of stabilizers of cells).
Both points of view are useful.Note that in contrast to the situation with Coxeter complexes, the walls in the Moussong complex are not subcomplexes.

Midplanes and blocks in cells
Let (W J , J) be a finite Coxeter group and V J the Euclidean vector space on which W J acts.We summarize here the basic properties of a Coxeter complex of W = W J .For more about them see [Hum90] or [Bro96].We define a reflection in W J to be a conjugate of element of J.The reflecting hyperplanes H w of reflection w ∈ W J cut V J into polyhedral pieces, which turn out to be cones over simplices.In this way one obtains a simplicial complex C = C(W ) which triangulates the unit sphere in V J .This is called the Coxeter complex associated with W J .The group W J acts simplicially on C and this action is simply transitive on the set of maximal simplices (=chambers).Moreover the closure of any chamber C is a fundamental domain of the action of W on C, i.e., each x ∈ V is conjugated under W to one and only one point in C. Two chambers are adjacent if they have a common codimension one face.For any two adjacent chambers there is a unique reflection in W J interchanging these two chambers.
A similar picture we have for the Coxeter cell X J .By a midplane in X J we mean the intersection H w ∩ X J , where w ∈ W J is a reflection and H w its reflecting hyperplane.We denote this midplane by M (J, w).By equivariance we define the notion of a midplane in any cell of M W .Each midplane M defines a unique cell in M W , the cell of least dimension in M W which contains M , and we will denote this by C(M ).
Lemma 2.1.1Every cell X J contains an open neighborhood of the origin of V J .
In particular midplanes in X J have dimension |J| − 1 and there is one-to-one correspondence between reflecting hyperplanes and midplanes.
Proof.Note first that the ray R + x J lies in the interior of the chamber C = {x ∈ V J : B(x, e s ) > 0 ∀s ∈ S}.Hence in each chamber wC, w ∈ W J there is a vertex wx J of X J .Now suppose that X J does not contain the origin in the interior, then there is a hyperplane H through the origin such that X J is contained in one of the closed half-spaces defined by  Lemma 2.1.3The only faces of a cell X K having nonempty intersection with midplane M (K, s), s ∈ S are those wX JK with w −1 sw ∈ W J .In particular M (K, s) contains no vertices of X K .More generally a face of X K has nonempty intersection with midplane M (K, usu −1 ), s ∈ S, u ∈ W iff it is of the form uwX JK with w −1 sw ∈ W J .
Proof.If w −1 sw ∈ W J then swW J = wW J , that is s leaves the vertex set of wX JK invariant and hence it leaves invariant the cell itself and has a nonempty fixed-point set in this cell.Conversely if M (K, s) ∩ wX JK is nonempty then there a face F of the cell wX JK such that M (K, s)∩F contains an interior point of F .But then s leaves F invariant hence by Lemma 1.4.2 it also leaves any "overcell" invariant in particular wX JK and this implies that w −1 sw ∈ W J .
To deduce the second statement from the first, one need only to note that M (K, wsw −1 ) ∩ wX JK = w(M (K, s) ∩ X JK ).

G. A. Noskov
Lemma 2.1.4If w ∈ W J leaves invariant some midplane M in X J then it fixes this midplane pointwise.
Proof.Indeed, w leaves invariant the ambient face C and we can apply Lemma 1.4.1.
Lemma 2.1.1For any cell X K the following hold: 1.The intersection of a midplane of X K with any of its face is again a midplane.
2. Any midplane of any face of X K is an intersection with this face of a precisely one midplane of X K .
Proof. 1) We may assume that a given midplane M is of the form M (K, s) and the face of X K is X JK , J ⊆ K. Since s belongs to W J , it leaves X JK invariant and its fixed-point set X s JK bijects onto the fixed-point set X s J by a W J −equivariant isometry p J |X JK : X JK → X J .The general assertion follows by equivariance.
2) We may assume that the face is of the form X JK for J ⊆ K. Let M JK be a midplane of X JK , then by definition is an edge of the face X JK , flipped by w.The intersection M JK ∩ σ = {m} is a midpoint of σ and M JK is orthogonal to σ.Now if M any midplane with the same intersection with X JK as M K , then the reflection in M flips the edge σ and hence this edge is orthogonal to M and thus M = M K .
Lemma 2.1.51)For every x ∈ M (K, s) ∩ X JK there is a nondegenerate segment of the form [y, sy], y ∈ X JK with x as a midpoint.2) The segment [y, sy] is orthogonal to midplane M (K, s). 3) For any midplane M in X K there is an edge of X K , intersected by M in the midpoint.
Proof. 1) Since M (K, s)∩X JK is nonempty, it follows from Lemma 2.1.3that s ∈ J. Let U = {u ∈ W J ; x k and ux K are on the same side of M (K, s).} Clearly W J = U ∪ sU, U ∩ sU = ∅ and the sets U x K , sU x K lie entirely on the different sides of the midplane M (K, s).Since X JK = Ch(W J x K ), we have where u∈U (λ u + µ u ) = 1 and all coefficients are nonnegative.Since x is fixed by s, applying s to both parts of the equality above we get We conclude from these two equalities that x = 1/2(y + sy), where y = 2) The segment [y, sy] is orthogonal to M (K, s) since it is flipped by an orthogonal transformation s.
We will call the segment [y, sy] from the lemma above to be a perpendicular to Proof.It follows from the fact that the tangent space in x is orthogonal sum of a the tangent space of M (K, s) and a tangent space of the segment [y, sy].

Walls as equivalence classes of midplanes
We assume that M = M W is the Moussong complex of a Coxeter group W .The following definition mimics the definition of a hyperplane in a cube complex given in [NR98].
Definitions 2.2.1 For midplanes M 1 and M 2 of the cells C 1 = C(M 1 ) and is again a midplane (and then of course it is a midplane of C 1 ∩ C 2 ).The transitive closure of this symmetric relation is an equivalence relation, and the union of all midplanes in an equivalence class is called a wall in M. Clearly the equivalences above are generated by those of the form Thus to prove some property P for midplanes of a wall H it is enough to prove this property for some midplane in H and then show that the validity of P is preserved under equivalences just mentioned.If M is a midpoint of a 1-cell(=edge) in M then the wall spanned by M will be called a dual wall of e and denoted by H(e).We denote by H M the union of midplanes in the equivalence class of a midplane M.
It follows immediately from Lemma 2.1.5that Lemma 2.2.2Any wall H of M has the form H(e) for some edge e.
Clearly W acts on the set of midplanes, preserving the equivalence relation and hence acts on the set of walls.For any wall H we denote by H the complex obtained from the disjoint union of midplanes in H by gluing any two midplanes in H along their common submidplane in M (if such one exists).One can easily see that H is nonpositively curved, i. e. satisfies the link condition.Namely, the link of any cell C of M is isometric to the product C × [−π, π].

G. A. Noskov
Lemma 2.2.3Let p : H → M be the natural map which sends each midplane in H to its image in M. Then p is an isometry of H onto H.As a consequence of the above walls are convex in M.
Proof.It is similar to the proof of lemma 2.6 in [NR98].Clearly, p is an isometry on each midplane.By result of M. Gromov ([Gro87], Section 4) it is enough to show that p is a local isometry, that is if x ∈ H, then there is a neighborhood U of x such that p| U is an isometry.Clearly p bijects the star St(x) onto the union U of all midplanes, containing p(x).This union is the fixed-point set of some reflection from W ( see Lemmas 1.4.1, 1.4.2,1.4.4).Hence U is convex, and p maps St(x) isometrically onto U.
Lemma 2.2.4Each wall in M W is the fixed-point set of a precisely one reflection in W . Conversely, the fixed-point set of a reflection in W is a wall.
Proof.Let H M be the wall, spanned by a midplane M of the cell C. From the description of cells and that of the action of W we know that M is the fixed point set of a reflection from the stabilizer S C of C in W. We will show that H M coincides with the fixed-point set H w of w.
Any reflection w fixing a midplane M pointwise fixes also H M pointwise, i.e., H M ⊆ H w .We have to show that the claimed property is invariant under equivalence relation of midplanes, see Every wall H is the fixed-point set of a unique reflection in W. Write H as the dual wall H = H(e) of some edge e of M. If there were two reflections w, w ′ with the same reflection wall H then their difference w −1 w ′ would fix e pointwise.But W acts simply transitively on the vertices of M hence w = w ′ .Now any w ∈ W fixing at least one cell pointwise is an identity.Indeed the set of cells fixing by w pointwise is nonempty and containing with each cell C every its "overcell" C ′ ⊃ C because by Lemma 1.4.2wC ′ = C ′ and since the stabilizer of C ′ acts fixed point free on the cell we conclude that w = 1.
H w coincides with H M .Suppose, to the contrary, that there is a w-fixed point x outside H M .Take any y ∈ H M , then w fixes the endpoints x, y of the geodesic [x, y] hence, by uniqueness, it fixes the whole geodesic.Shortening [x, y] if necessary we may assume that [x, y) is outside H M .Take z ∈ [x, y], z = y such that the open segment (z, y) is contained entirely in the interior of some cell C. Since w fixes (z, y), it leaves C invariant.As far as y ∈ H M ∩ C, the point y is contained in some midplane M ′ ⊂ H M of C. Because w fixes M ′ and the segment (z, y), lying entirely outside H M , we conclude that w fixes C pointwise -contradiction.
For the converse, let w be a reflection in W. Note first that H w contains at least one midplane.Indeed, since any reflection in W is conjugate to some s, s ∈ Documenta Mathematica 16 (2011) 373-398 S, we may assume that w = s.Take J = {s}, then the cell X J = Ch(x J , sx J ) is a segment on which s acts as a reflection thereby fixing its midpoint M .We conclude that H w contains H M for some midplane M .Therefore, as was proved above, H w coincides with H M .Lemma 2.2.5 The edge path in M (1) is geodesic if and only if it crosses each wall at most once.
Proof.If an edge path p = e 1 e 2 • • • e k crosses a wall H twice, say distinct edges e i , e j , i < j cross H, then we delete the subpath e i • • • e j and instead insert the path w(e i+1 • • • e j−1 ), where w is the reflection in the wall H.The resulting path is strictly shorter than p but connects the same vertices.Conversely, suppose that an edge path p from x to y crosses each wall at most once.Let H H be the set of all walls crossing by p. Since x and y are at the different sides of each wall from H H , we conclude that any path from x to y should cross than that of p.Any wall in the Moussong complex is "totally geodesic" in the following sense Lemma 2.2.6 Any geodesic in M having nondegenerate piece in a wall H, lies entirely in H.

Separation properties
Lemma 2.3.1 Every wall in M separates M into exactly two connected components.
Proof.First, we claim that H separates M into at least two components.We know from Lemma 2.2.2 that H = H(e) -the dual wall of some edge e = [x, y].We will show that x, y belong to different connectedness components.Suppose, to the contrary, that x, y are in the same connectedness component.Then

G. A. Noskov
there is a closed edge path α in M (1) crossing H only once.(Clearly any edge either intersects H in a midpoint or does not intersects H at all.)Since M is contractible this path can be contracted to a constant path by a sequence of combinatorial contractions in cells.By Lemma 2.1.1 any cell C either has an empty intersection with H M or H M ∩ C is a midplane of C.This implies that each combinatorial contraction of the edge path in the cell does not change the number of intersections with H M modulo 2. Since this number is 0 for the final constant path, it cannot be 1 for the initial path.To prove that the H cuts out M into exactly two components, we proceed as in [NR98], lemma 2.3 (preprint version.)Notice first that H is 2-sided, that is there exists a neighborhood of H in M which is homeomorphic H ×I, I = [0, 1].Indeed, by Lemma 2.1.5, in each cell there is a neighborhood which is fibered as M × I: the fibrations can be chosen to agree on face maps so this induces an I-bundle structure on some neighborhood N over H. Since H itself is CAT(0) it is contractible so the bundle is trivial.It follows that N has two disjoint components, {−1/2} × H and {1/2} × H. Any point in the complement of H can be joined to one of these boundary components by a path in the complement of H, and therefore X − H has exactly 2 components as required.
Lemma 2.3.1 For any wall H both components of the complement M − H are convex.
Proof.Suppose that x 1 , x 2 lie on the same side of H, say H + .We claim that [x 1 , x 2 ] lies entirely in H + .Suppose the contrary, then by Lemma 2.2.6 the intersection [x 1 , x 2 ] ∩ H consists of precisely one point, say x.Similar to the proof of Lemma 2.2.6 there are segments σ 5) The interiors of σ 1 , σ 2 are contained entirely in H + .Then it follows from Lemma 2.1.5that there exists a reflection w ∈ W and a segment [y, wy] such that the segment has x as a midpoint and is orthogonal to both M 1 and M 2 .By interchanging the roles of y and wy if necessary we may assume that y ∈ H + .Denote σ = [x, y], σ ′ = [x, wy].It follows from Lemma 2.1.6that the angles ∠ x (σ 2 , σ), ∠ x (σ 2 , σ ′ ) are both strictly less than π/2.But a small nonzero move of x along σ would strictly shorten the length of σ 1 ∪ σ 2 contradicting the assumption 4) above.

Chambers
Since the complex M is locally finite and there are only finite number of midplanes in each cell, we conclude that the set of all walls H in M is locally finite, in the sense that every point of M has a neighborhood which meets only finitely many H ∈ H. Definition 3.1.1By Lemma 2.3.1 the walls H ∈ H yield a partition of M into open convex sets, which are the connected components of the complement M − (∪ H H). We call these sets chambers.
To distinguish chambers from cells, we will denote them by letter D, possibly with indices, dashes, etc. Lemma 3.1.2For any two distinct chambers D(x), D(y), x, y ∈ M (0) there is a wall H separating them.
Proof.Consider a geodesic edge path p = e 1 e 2 • • • e k from x to y, then by Lemma 2.2.5 H(e 1 ) separates x from y and hence separates D(x) from D(y).
Lemma 3.1.3Each chamber contains precisely one vertex of M.
Proof.Since W acts simply transitively on the set of vertices of M and each vertex is contained in some chamber we deduce that each chamber contains at least one vertex.Now, if x, y are distinct vertices in a chamber C, we connect them by a geodesic path p in M (1) .Then by criterion of geodesicity any wall crossed by p separates x from y, contradicting the definition of chamber.
In view of this lemma we will write D(x) for the chamber containing the vertex x of M. Lemma 3.1.51) Every chamber is uniquely determined by any of its maximal blocks.2) Every chamber is a union of maximal blocks, and it contains at most one maximal block from each maximal cell.

G. A. Noskov
Proof. 1) Indeed, the interior of a maximal block is open in M and does not intersect any wall, consequently there is only one chamber containing this block.
2) Since M is a union of maximal cells, any chamber is a union of maximal blocks.Take a chamber D, then Proof.Suppose, to the contrary, that there is b ∈ D∩D ′ which is not contained in H. Since H is closed a small neighborhood of b does not intersect H.But this neighborhood contains points both from D and D ′ , which thus belong to one halfspace of H, contradicting the separation hypothesis.Lemma 3.1.8Two distinct chambers D(x), D(y) (x, y ∈ M (0) ) are adjacent if and only if the vertices x, y are adjacent in M (1) .For any two adjacent chambers there is a reflection in W , permuting these chambers and fixing the intersection of their closures pointwise.
Proof.The lemma is about Coxeter cell, thus it follows from the description of its structure as a Coxeter complex.Proof.Since D = ∩{H s − : s ∈ S} contains x 0 , it contains also D 0 .Let B J be a block of a maximal cell X J , containing x J = x 0 .Then B J ⊂ D 0 -indeed it follows from the description of the chambers in the Coxeter complex that B J is bounded by the hyperplanes H s = e ⊥ s , s J. Suppose now that D strictly contains D 0 and let x ∈ D − D 0 .Since D is convex, the whole segment [x, x 0 ] lies in D. Let T = (x 0 , x 1 , . . ., x m ) be a taut chain from x 0 to x m = x.The first piece [x 0 , x 1 ] lies entirely in some maximal cell of the form X K and we know that the block B K = D 0 ∩ X K is the maximal block in X K and it is bounded by the hyperplanes is not a vertex of X K , then x 1 is the boundary point of X K and hence it is contained in the interior of some face F of X K .If F contains x 0 , then all three points x 0 , x 1 , x 2 lie in some cell contradicting to the choice.Hence F does not contain x 0 and thus the open interval (x 0 , x 1 ) lies entirely in the interior of X J and hence crosses some wall H s , s ∈ J -contradiction.

Galleries Definitions
Recall that the chambers are in one-to-one correspondence with the vertices of M and chambers are adjacent if and only if the correspondent vertices are adjacent in the 1-skeleton of M. It follows immediately that the following lemma is true.Lemma 3.2.2 1) Any two chambers D, D ′ can be connected by a gallery of length d(D, D ′ ).2) A gallery is geodesic if and only if and only if it does not cross any wall more than once.3) Given s 1 , . . ., s d ∈ S, there is a gallery of the form D 0 (s Conversely, any gallery starting at C has this form.4) The action of W is simply transitive on the set of chambers.

Lemma 3.2.3
There is a constant c(M) such that for any two distinct chambers D, D ′ with nonempty intersection D ∩ D ′ , there is a geodesic gallery Proof.Let H 0 be the set of walls separating D from D ′ .In view of Lemma 3.2.2, it is enough to bound the cardinality of H 0 .According to Lemma 3.1.7each Clearly the number of cells containing x is uniformly bounded and for each such a cell C the number of midplanes in C containing x is also uniformly bounded.Since a wall is uniquely determined by any of its midplanes, this proves the lemma.

Approximation property
Definition 3.3.1 Let (X, G) be a pair consisting of a geodesic metric space X and a graph G embedded into X.We say that (X, G) satisfies the approximation property if X-geodesics between the vertices of G can be uniformly approximated by geodesics in G.This means that there is a constant δ such that for any X-geodesic α X between the vertices of G there is a G-geodesic α G between the same vertices such that α X and α G lie entirely in the δ-neighborhoods of each other.We will express this by saying that α X , α G are δ-close to each other.
Of particular interest is the case when G is the embedded Cayley graph of a group acting on X.
Theorem 3.3.2Let (W, S) be a Coxeter group and let M be its Moussong complex.Embed the Cayley graph C W as an orbit W x 0 for a point x 0 in a base chamber D 0 of M. Then the pair (M W , C W ) satisfies the approximation property.
Proof.Let σ = [a, b] be a nondegenerate segment in M and H σ be the set of walls having a nonempty intersection with the interior (a, b).Since the family of all walls is locally finite and the walls are totally geodesic, we have a partition where the walls from H ′ σ contain σ and the walls from H i cross σ precisely in the point a i , i = 1, • • • , n, and a = a 0 < a 1 < • • • < a n < a n+1 = b.Now we define a gallery Γ along the geodesic σ = [a, b] as the gallery is a geodesic gallery and the lengths of spherical pieces are bounded from above by the constant c(M) from Lemma 3.2.3, 3) Each spherical piece D i Γ i D i+1 , i = 1, . . ., n crosses the walls only from the set H ′ σ ∪ H i .
Lemma 3.3.3For any geodesic σ = [a, b] in M there is a geodesic gallery along σ.
Proof of the lemma.By construction of the sequence {a i }, for each i = 1, . . ., n + 1 there is a chamber is the first approximation to the required gallery.In general, this sequence is not a gallery, since two consecutive chambers are not necessarily adjacent.For each 1 ≤ i ≤ n, the intersection of neighbors D i ∩ D i+1 contains the point a i .

Documenta Mathematica 16 (2011) 373-398
Application of Lemmas 3.1.7,3.2.3enables us to inscribe a spherical geodesic subgallery of bounded length between these neighbors and get a gallery such that the spherical pieces D i Γ i D i+1 are geodesic galleries of uniformly bounded length satisfying condition 3) from the definition above.We will show that Γ can be modified to a geodesic gallery along  Let G be a group with a fixed word metric x → ℓ(x).We say that an element x = 0 is straight if ℓ(v n ) = nℓ(v) for all natural n.Recall that M is a proper complete CAT(0) space and W acts properly and cocompactly on M by isometries.In particular, any element w ∈ W of infinite order acts as an axial isometry i.e., there is a geodesic axis A w in M, isometrical to R, on which w acts as a nonzero translation [Bal95].We say that w is generic if A w intersects any wall in at most one point.In view of Lemma 2.2.6, this is equivalent to saying that no nondegenerate segment of A w is contained in a wall.
Theorem 4.1.5Let (W, S) be a Coxeter system of finite type.For any generic element w of W of infinite order there is a conjugate v which is straight, that is ℓ(v n ) = nℓ(v) for all n ∈ N, where ℓ(v) is a word length in generators S.
Proof.We make use of the action of W on the Moussong complex M. Since the family of all walls is locally finite, there is a point a on the axis A w such that a does not belong to any wall of M. Every point w i a(i ∈ Z) also does not belong to any wall of M. Let H be the set of walls crossed by the segment [a, wa] and let a < a 1 < a 2 < • • • < a k < wa be the crossing points, so that H is a disjoint union ) is a gallery, crossing only the walls from H and crossing each wall precisely once.In particular this gallery is geodesic.
Translating by w and concatenating, we get a gallery Γ = Γ 0 (wΓ 0 ))(w The walls that it crosses are precisely those from the union H ∪ wH ∪ w 2 H ∪ • • • ∪ w n−1 H, and each wall is crossed precisely once.Hence the gallery Γ is geodesic.Now let D 1 = uD 0 , u ∈ W, where D 0 is the base chamber.Being a geodesic path in the Cayley graph, the gallery Γ joins the vertex u to the vertex w n u = u(u −1 w n u).
Hence its length nℓ(Γ 0 ) equals the word length of the element u −1 w n u ∈ W.
Theorem 4.1.6Let (W, S) be a Coxeter group of finite type.There is a constant c = c(W ) such that for any element w of W of infinite order there is a conjugate v of w c which is straight.
Proof.Let w ∈ W be of infinite order and let A w be an axis of w.Let of the length nℓ(Γ).This gallery crosses the walls only from (disjoint) union H ∪ w c H ∪ w 2c H ∪ • • • ∪ w (n−1)c H, each precisely once.Hence the gallery Γ is geodesic.Now let D = uD 0 , u ∈ W, where D 0 is the base chamber.Being a geodesic path in the Cayley graph, the gallery Γ joins the vertex u to the vertex w nc u = u(u −1 w nc u).Hence its length nℓ(Γ) equals the word length of the element u −1 w nc u ∈ W. We conclude that for v = u −1 w c u the equality |v n | = n|v| holds for all n ∈ N.
For elements which are not necessarily generic we have the following Lemma 4.1.7Let (W, S) be a Coxeter group of finite type and let w ∈ W be an element of infinite order.Fix an axis A w of w in the Moussong complex M W .There is a chamber D such that for all n ∈ Z where |c n | is bounded by a constant depending only on W and H w is the set of all walls H in M W , containing A w and such that H separates w i D from w i+1 D for some i ∈ Z.
Proof.We follow the proof of Theorem 4.1.5.Take a chamber D, such D ∩A w is a nondegenerate segment.Let H be the set of walls, separating D from wD and do not containing A w .By total geodesicity, any H ∈ H crosses A w precisely in one point.Let Γ be a geodesic gallery from D to wD then it crosses all H ∈ H, each precisely once, and some of the walls from H w .Iterating we get the gallery Γ = Γ(wΓ))(w 2 Γ) • • • (w n−1 Γ)w n D. This gallery crosses the walls from (disjoint) union H ∪ wH ∪ w 2 H ∪ • • • ∪ w n−1 H, each precisely once.Also, it crosses some walls from H w .Note that, whenever Γ crosses H ∈ H w , it crosses it periodically with a period r H = card w Z H. Hence, the integer part [n/r H ] is the number of times the gallery Γ crosses each H ′ ∈ w Z H. Hence it crosses the walls from the orbit w Z H approximately n times, up to a universal constant.Hence, the number d(D, w n D) of walls, separating D from w n D, equals n d(D, wD) − n card (w Z \H w ) + c n , where c n is uniformly bounded.
From this we get the following formula for a translation length ||w|| of w: In particular, translation length of any element of W is rational (even integral).
Remark 4.1.9The formula for translation length is similar to the one given in [Kra94], where it follows from the classification of roots.It seems unknown whether translation length is rational in an arbitrary "semihyperbolic group".

Norms and Burago's inequality
Let A be a normed abelian group, so A is equipped with a function ℓ : In general positivity of Rℓ fails, so it is possible that Rℓ(a) = 0 but a = 0. Also it may easily happen that Rℓ is not Hausdorff equivalent to ℓ.We give a criterion for positivity and Hausdorff equivalence in terms of Burago's inequality [Gro93].
Definition 4.2.2We say that a norm ℓ on an abelian group A satisfies the Burago's inequality if there exists a constant c = c(A) > 0 such that ℓ(a 2 ) ≥ 2ℓ(a) − c for all a ∈ A.
The norm is discrete if for all n ∈ N the ball B n = {x ∈ A : ℓ(x) ≤ n} is finite.For example any word metric, corresponding to a finite generating set, is discrete.Lemma 4.3.1 Let Γ be a finitely generated group of isometries of a proper CAT(0) space X, acting cocompactly and properly on X. Suppose that x 0 ∈ X has a trivial stabilizer so that the Cayley graph C of Γ can be considered as embedded into X via the orbit map g → gx 0 (g ∈ Γ).Suppose that the pair (X, Γx 0 ) satisfies the approximation property.Then the restriction of the word length ℓ on Γ to any finitely generated free abelian subgroup A satisfies the Burago's inequality.
Proof.By assumption there is a δ > 0 such that for any g ∈ Γ the X-geodesic α X from x 0 to gx 0 and some C-geodesic α C from x 0 to gx 0 are δ-close to each other.By the flat torus theorem [Bow95], [Bri95] there is a Euclidean subspace F in X on which A acts by translation.Fix the point y 0 ∈ F and let a be an arbitrary nontrivial element in A. We will show that ax 0 is contained in a c-neighborhood of α C for a suitable c > 0. Clearly d X (a 2 x 0 , a 2 y 0 ) = d X (x 0 , y 0 ).Parameterize the segments [x 0 , a 2 x 0 ], [y 0 , a 2 y 0 ] by the segment [0, 1] proportionally to arc length.It follows from the convexity of X-metric that the corresponding points on the segments are distance at most d X (x 0 , y 0 ) from each other.Let u be the point on [x 0 , a 2 x 0 ] corresponding to the point ay 0 .By assumption u is distance at most δ from some point v on α C .Hence we have bounded the X-distance from ax 0 ∈ C to v ∈ C. (This key observation is illustrated in Figure 4).Since the Cayley graph C is quasiisometric to X this bounds the Cayley graph distance also.Thus, there is a constant c = c(Γ, X) > 0 such that d C (ax 0 , v) ≤ c.We have ℓ(a 2 ) = d C (x 0 , v)+d C (v, a 2  I am grateful to Herbert Abels for asking the question that leads to the theorem above. For any positive ǫ < d C (a, b)/2 we may choose points x and y in the interior of α such that d(a, x) = ǫ = d(y, b) A path from x to y obtained by traveling along α to a then along γ to b has length length(γ) + 2ǫ, while a geodesic from x to y has length d C (a, b) − 2ǫ, so d C (a, b) ≤ length(γ) + 4ǫ.Since this is true for any sufficiently small ǫ > 0, we conclude that d C (a, b) ≤ length(γ), and so α is a geodesic from a to b.
and w fixes M 1 then w leaves C 1 invariant, hence by Lemma 1.4.2 it leaves C 2 invariant and by Lemma 2.1.1 it leaves M 2 invariant and finally by Lemma 2.1.4it fixes M 2 pointwise.In case M 1 ∼ M 2 , C 1 ≥ C 2 , and w fixes M 1 pointwise it is clear that w fixes M 2 pointwise.
Definitions 3.1.4Recall from §2.1.2that midplanes of any cell C in M yield a partition of C into convex (open) blocks.(Blocks are open in C, not in M.) A maximal block is a block in a maximal cell.Two maximal blocks are adjacent if they are contained in the same maximal cell and share a codimension one face.Two chambers D, D ′ are adjacent if there are maximal blocks B ⊂ D, B ′ ⊂ D ′ which are adjacent.A wall H is a wall of a chamber D if there is a maximal cell C such that H ∩ C contains a codimension one face F a maximal block B of D.
The intersection D ∩ C is a union of maximal blocks because D ∩ C is an intersection of open half-cells in M .Next, if D contains two maximal blocks B 1 , B 2 from one cell, then there is a midplane M separating B 1 from B 2 and the ambient wall H also separates B 1 from B 2 contradicting the definition of D.Lemma 3.1.6Let B, B ′ be maximal adjacent blocks and let D, D ′ be corresponding ambient chambers.Let H be a wall separating B from B ′ .Then H is the only wall that separates D from D ′ .Proof.Let C a maximal cell containing B, B ′ , then B, B ′ are adjacent in this cell and clearly there is only one midplane separating them.But the wall is uniquely determined by any of its midplanes, whence the lemma.Lemma 3.1.7Let D, D ′ be chambers such that their closures D, D ′ have a nonempty intersection.Let H be a wall, separating D from D ′ .Then H contains the intersection D ∩ D ′ .
Definition 3.1.9The base chamber D 0 of M is the chamber, containing the base vertex x 0 of M. For each s from the generating set S of W, we denote by H − s those open halfspace of the wall H s , which contains the base vertex x 0 .Lemma 3.1.10D 0 = ∩{H s − : s ∈ S}.Documenta Mathematica 16 (2011) 373-398
Theorem 4.1.8If, under conditions of Lemma 4.1.7,D = uD 0 , u ∈ W, where D 0 is the base chamber, then d(D, wD) is the word length of the conjugate v = u −1 wu ∈ W and we get the following formula ℓ 2) ℓ(ab) ≤ ℓ(a) + ℓ(b), and (3) ℓ(a) ≥ 0 with ℓ(a) = 0 iff a = 1, for a, b ∈ A. If (3) is replaced by (3') ℓ(a) ≥ 0 for a ∈ A, we call A a pseudonormed abelian group.Two pseudonorms ℓ and ℓ ′ on the abelian group A are called Hausdorff equivalent if there is a constant k > 0 so that |ℓ(a) − ℓ ′ (a)| ≤ k for all a ∈ A. The (pseudo)norm ℓ on the abelian group A is called regular if ℓ(a n ) = nℓ(a) for all a ∈ A and all positive natural numbers n.Let ℓ be a norm on the abelian group A. We define the regularization Rℓ of ℓ by Rℓ(a) = lim n→∞ ℓ(a n ) n .By [PS78], p. 23, Exercise 99, this limit always exists, and it is an exercise to see that Rℓ is a regular pseudonorm.Documenta Mathematica 16 (2011) 373-398 Lemma 4.2.1 The norm ℓ on the abelian group A is regular iff Rℓ = ℓ.Proof.If ℓ is regular, then clearly Rℓ = ℓ.Conversely, if ℓ(a n ) < nℓ(a) for some positive number n and some a ∈ A, then Rℓ(a) = lim m→∞ ℓ(a mn ) mn ≤ ℓ(a n ) n < ℓ(a), thus the lemma.

Figure 4 :
Figure 4: Lemma 4.3.1 Coxeter complex on E. Let, for example, W be an affine Coxeter group generated by the reflections s 1 , s 2 , s 3 in the sides of an equilateral triangle C in the Euclidean plane.Then M W is the tessellation of the plane by hexagons, dual to the tessellation consisting of the images of C us cover β by intervals of radius ǫ/2.Each such an interval is geodesic.Indeed, a geodesic γ connecting the points of the interval can not cross ∂C, hence it lies in the union U of cells, having C as a face.For any cell C ′ , having C as a face, γ can not cross ∂C ′ − C since it has to pass a distance at least ǫ.Hence it lies in only one such C ′ and thus coincides with the interval.It follows from the considerations above that β is a local geodesic, and therefore a global geodesic since X is CAT(0).Now let γ be a path in X joining a to b.
Two blocks are adjacent if they have a common codimension one face.There is a canonical one-to-one correspondence between blocks in X J , chambers of the Coxeter complex C(W J ) and vertices of X J .This correspondence clearly preserves the adjacency relation.Each block contains a unique vertex of X J since a closed block B is a fundamental domain of the action of W on X J , i.e., each x ∈ X J is conjugated under W to one and only one point in B. The group W J acts on the set of blocks and this action is simply transitive.For a block B the intersection of the closed block B with a midplane is called by internal face of B.
H, say in H + .This implies that each chamber has an interior point, lying in H + .Take an arbitrary closed chamber D. If D lies entirely in H + then −D lies in the opposite half-space H − and hence there is no interior point in it belonging H + -contradiction.If D does not lie entirely in H + then H separates some codimension one face F of D from the remaining vertex x of D. Let D ′ be the chamber, adjacent to Documenta Mathematica 16 (2011) 373-398 D in a face F , then D ′ lies entirely either in H + or in H − and the previous argument works.Definitions 2.1.2It follows from Lemma 2.1.1 that the midplanes M (J, w) also cut X J into (relatively open) polyhedral pieces of dimension |J| -blocks.
[a, b].If Γ is not geodesic then by Lemma 3.2.2 it crosses some wall H at least twice.Clearly H ∈ H ′ σ i.e., H contains σ.Then there are indices i + 1 < j and subgalleries Γ 1 , Γ 2 each of length 1 such that a) Γ 1 , Γ 2 belong to i-th and j−th spherical piece respectively, b) Γ 1 , Γ 2 cross H and moreover there are no crossing subgalleries in between.Let Γ 1 = DD ′ , Γ 2 = D ′′ D ′′′ .In particular the chambers D and D ′′′ lie on the same side of H, say H − , and the subgallery Γ ′ of Γ, joining D ′ with D ′′ lies on the opposite side, say H + .Let w ∈ W be the reflection in the wall H.
(An element that is not straight.)Let W be an affine Coxeter group generated by reflections s 1 , s 2 , s 3 in the sides of an equilateral triangle C of a Euclidean plane.Let L 1 , L 2 , L 3 be the corresponding reflecting lines of this triangle.It is easily seen that there is nontrivial translation u ∈ W with an axis L 1 .We assert that nor s 1 u neither any of it conjugates v = ws 1 uw −1 are straight.Indeed, the length |ws 1 uw −1 | is the length of a geodesic gallery Γ from C to ws 1 uw −1 C. Any such a gallery intersects the line wL 1 .The concatenation Γ(vΓ) is a gallery from C to v 2 C of length 2|ws 1 uw −1 |.But Γ(vΓ) can not be geodesic, since it intersects wL 1 twice.Hence |v 2 | < 2|v|.Let M be the Moussong complex of a Coxeter group W .
denote the set of walls in the Moussong complex M W , containing A u .It is easy to see that the cardinality of H w is bounded by a constant depending only on W and we take c = c(W ) to be the number2 × l.c.m. × (card{H w : w ∈ W is of infinite order }).Clearly A w is an axis of w c as well.Furthermore, w c leaves invariant each wall H ∈ H w ; moreover, it leaves invariant each of the two components of M W − H, H ∈ H w .It follows that for any chamber D, a geodesic gallery from D to w c D does not cross a wall H from H w .Indeed, otherwise D and w c D would lie in different components of M W − H implying that w c interchanges these components, contradicting the property above.Take a chamber D such that D ∩ A w is a nondegenerate segment and fix a point a in the interior of this segment.Let H denote the set of walls H that are crossed by the segment [a, w c a] but do not contain it.Clearly any H ∈ H separates D from w c D. And conversely, if H separates, then the points a, w c a lie in different components of M W − H implying that H crosses the segment [a, w c a] in precisely one point.Let Γ be a geodesic gallery from D to w c D then the walls that it crosses are precisely those from H, and each wall H ∈ H is crossed by Γ precisely once.Iterating we obtain a gallery Γ