A generalization of the Davis-Moussong complex for Dyer groups

A common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. Matthew Dyer introduced a class of groups, which we call Dyer groups, sharing this feature. This class includes, but is not limited to, Coxeter groups, right-angled Artin groups, and graph products of cyclic groups. We introduce Dyer groups by giving their standard presentation and show that they are finite index subgroups of Coxeter groups. We then introduce a piecewise Euclidean cell complex $\Sigma$ which generalizes the Davis-Moussong complex and the Salvetti complex. The construction of $\Sigma$ uses simple categories without loops and complexes of groups. We conclude by proving that the cell complex $\Sigma$ is CAT(0).


Introduction
There is extensive literature on Coxeter groups as well as on right-angled Artin groups and more generally graph products of cyclic groups.One common feature of these two families of groups is their solution to the word problem.It was given by Tits for Coxeter groups [Tit69] and by Green for graph products of cyclic groups [Gre90].The algorithm does not only give a solution to the word problem but also allows to detect whether a word is reduced or not.In his study of reflection subgroups of Coxeter groups, Dyer introduces a family of groups which contains both Coxeter groups and graph products of cyclic groups.A close study of [Dye90] also implies that this class of groups, which we call Dyer groups, has the same solution to the word problem as Coxeter groups and graph products of cyclic groups.A complete and explicit proof is given in [PS22] .
Similarly to Coxeter groups and right-angled Artin groups, the presentation of a Dyer group can be encoded in a graph.Consider a simplicial graph Γ with set of vertices V = V (Γ) and set of edges E = E(Γ), a vertex labeling f : V → N ≥2 ∪{∞} and an edge labeling m : E → N ≥2 .We say that the triple (Γ, f, m) is a Dyer graph if for every edge e = {v, w} with f (v) ≥ 3 we have m(e) = 2.The associated Dyer group D = D(Γ, f, m) is given by the following presentation It is natural to ask the following question.Consider a property P satisfied by both Coxeter groups and graph products of cyclic groups.Do Dyer groups also satisfy P?
In [DJ00] Davis and Januszkiewicz show that right-angled Artin groups are finite index subgroups of right-angled Coxeter groups.For a right-angled Artin group A they give right-angled Coxeter groups W and W ′ where W ′ and A are both finite index subgroups of W and moreover the cubical complexes corresponding to A and W ′ are identical.Inspired by this work we consider the following question: Are Dyer groups finite index subgroups of Coxeter groups?Out of a Dyer graph (Γ, f, m) we build a labeled simplicial graph Λ and prove the following statement.
The next corollary is a direct consequence.
Corollary 1.2 (Corollary 2.9).Every Dyer group is a finite index subgroup of some Coxeter group.
This has many interesting consequences, among others it implies that Dyer groups are CAT(0) [Dav08,Theorem 12.3.3],linear [Bou02, Corollary 2], and biautomatic [OP22].This is the starting point for a more precise study of their geometry.Coxeter groups are known to act geometrically by isometries on the Davis-Moussong complex, right-angled Artin groups are known to act geometrically by isometries on the Salvetti complex.Moreover graph products of cyclic groups are known to be CAT(0) by [Gen17,Theorem 8.20].The aim is to construct an analog of the Davis-Moussong and Salvetti complexes for Dyer groups and by way of the construction give a unified description of them.The piecewise Euclidean cell complex Σ associated to a Dyer group D is constructed as follows.One considers a simple category without loops X and a complex of groups D(X ).The development C of D(X ) will then encode the necessary information to build Σ.In Section 4.1, this is done for the case of spherical Dyer groups, where a Dyer group D is spherical if it decomposes as a direct product D 2 × D ∞ × D p with D 2 a finite Coxeter group, D ∞ = Z n for some n ∈ N and D p a direct product of finite cyclic groups.In Section 4.2, the construction of Section 4.1 is extended to any Dyer group.The complexes D(X ) and C are analogues to the poset of spherical subsets S and the poset of spherical cosets W S for Coxeter groups, which are recalled in Section 3.2.Finally Section 4.3 is devoted to the construction of the piecewise Euclidean cell complex Σ and culminates with the proof of the following statement.
As we do not assume the reader to be familiar with simple categories without loops (scwols), Section 3.1 recalls the definitions and statements needed for the construction of the scwol C. In Sections 3.2 and 3.3 we recall the constructions of the Davis-Moussong complex and of the Salvetti complex.
Remark 2.4.For a subset W ⊂ V , we can consider Γ W the full subgraph of Γ spanned by W and the restrictions f W = f W and m W = m E(Γ W ) .The triple (Γ W , f W , m W ) is again a Dyer graph.We denote the associated Dyer group by D W . From [Dye90], we know that that the homomorphism D W → D induced by the inclusion W ֒→ V is injective, hence D W can be regarded as a subgroup of D.
For i ∈ {2, p, ∞}, let Γ i be the full subgraph spanned by V i and D i be the Dyer group associated to the triple (Γ i , f V i , m V i ).Note that D 2 is a Coxeter group, D ∞ a right angled Artin group and D p a graph product of finite cyclic groups.We recall the definition of Coxeter groups.
Definition 2.7.Let Λ be a simplicial graph with set of vertices V = V (Λ) and set of edges E = E(Λ).Let m : E → N ≥2 be an edge labeling of Λ.The Coxeter group W = W (Λ) associated to the graph Λ is given by the following presentation span an edge in Λ if and only if they span an edge e = {u, v} in Γ, we set the label of the edge e = {u, v} ∈ E(Λ) to be m ′ (e) = m(e).For all u ∈ V p ∪ V ∞ and v ∈ V (Λ) \ {u, u ′ }, there is an edge e = {u ′ , v} ∈ E(Λ) labeled by m ′ (e) = 2. Finally for all u ∈ V p there is an edge e = {u, spans a complete graph in Λ.Let W = W (Λ) be the Coxeter group associated to the graph Λ.We give an action of (Z/2Z) |Vp∪V∞| on D.
We show that U is isomorphic to W by giving explicit homomorphisms φ : W → U and ψ : First consider the map φ : 5. Finally for every u ∈ V p there is an edge So the map φ extends to a homomorphism φ : We show that ψ extends to a homomorphism from U to W .

For all
If u ∈ V 2 and v ∈ V p ∪ V ∞ , we have m(e) = 2 and So the map ψ extends to a homomorphism ψ : U → W.
We now check that φ • ψ = Id U and ψ • φ = Id W by showing that these maps are the identity on the generators.For v ∈ V 2 , we have φ Corollary 2.9.Every Dyer group is a finite index subgroup of some Coxeter group.
Remark 2.10.As mentioned in the introduction, Corallary 2.9 has many interesting consequences.It implies that Dyer groups are CAT(0), linear, and biautomatic, that they satisfy the Baum-Connes conjecture, the Farrell-Jones conjecture, the Haagerup property and the strong Tits alternative.They also admit a proper and virtually special action on a CAT(0) cube complex.
Example 2.11.We apply the previous theorem to Example 2.6.The corresponding graph Λ is given in Figure 2.So by Theorem 2.8, the Dyer group D m,p is an index 4 subgroup of the Coxeter group Figure 2: The graph Λ m,p built out of the Dyer graph Γ m,p for some m, p ∈ N ≥2 .We color coded the vertices V ⊂ V (Λ) and For the edges: for edges of the form e = {u, u ′ } and for edges of the form Every edge is labeled by 2 if not specified otherwise.
The Dyer group D is not the only Dyer group which is a finite index subgroup of W .We describe such another Dyer group D ′ = D(Ω, g, n) by giving the Dyer graph (Ω, g, n).We define a second Dyer graph (Ω, g, n) starting from Γ.The vertices of Ω are both span copies of Γ, with same labeling of edges, and for u, v ∈ V ∞ the vertices v, u ′ span an edge labeled by 2 in Ω if and only if v and u span an edge in Γ.The labeling of the vertices is defined as follows g V 2 ∪V∞∪V ′ ∞ = 2 and g Vp = f Vp .Let D ′ be the Dyer group associated to Ω.Note that every generator x v , v ∈ V (Ω) of D ′ has finite order.We now give an action of (Z/2Z) |Vp∪V∞| on D for all e = {u, v} ∈ E(Ω) As in Theorem 2.8, we can check that U is isomorphic to W by considering explicit homomorphisms φ : W → U and ψ : The map φ : {y v , v ∈ V (Λ)} → U is given as follows: for u ∈ V 2 , φ(y u ) = x u , for u ∈ V p , φ(y u ) = κ u x u and φ(y u ′ ) = κ u and for u ∈ V ∞ , φ(y u ) = x u and φ(y u ′ ) = κ u .One can easily check, using methods which are similar to those used in the proof of Theorem 2.8, that the map φ induces a homomorphism φ : W → U.

The map ψ
for u ∈ V 2 , ψ(x u ) = y u and for u ∈ V p , ψ(x u ) = y u ′ y u and ψ(κ u ) = y u ′ , finally for u ∈ V ∞ , ψ(x u ) = y u , ψ(x u ′ ) = y u ′ y u y u ′ and φ(κ u ) = y u ′ .Again one can easily check, using methods which are similar to those used in the proof of Theorem 2.8, that the map ψ induces a homomorphism ψ : There are two types of vertices: Every vertex and every edge is labeled by 2 if not specified otherwise.
Example 2.13.We apply the previous theorem to Example 2.6.The corresponding graph Ω m,p is given in Figure 3.The associated Dyer group is It

Introduction to scwols
Small categories without loops (scwols) and complexes of groups were introduced by Haefliger in [Hae91], [Hae92].Based on [BH99], we would like to recall some notions about scwols and complexes of groups, as we do not assume the reader to be familiar with these constructions.We hope to give all necessary definitions and results, details can be found in [BH99].The reader familiar with scwols might only consider the specific examples developed in this section as they will be used in the construction of the cell complex Σ.
A small category without loops (scwol) X consists of a set V (X ), called the vertex set of X and a set E(X ), called the set of edges of X .Additionally two maps i : E(X ) → V (X ) and t : E(X ) → V (X ) are given.We call i(α) the initial vertex of α ∈ E(X ) and t(α) the terminal vertex of α ∈ E(X ).The set associates to each pair (α, β) an edge αβ, called their composition.These sets and maps need to satisfy the following conditions: Remark 3.1.To any poset (P, <) we can associate a scwol X where the set of vertices is P and the set of edges are pairs (a, b) ∈ P × P such that b < a, t((a, b)) = a and i((a, b)) = b.Definition 3.2 (Product of scwols).Given two scwols X and Y, their product X × Y is the scwol defined as follows: whenever defined.
Example 3.4.Consider a finite set S. Let P(S) be the set of subsets of S. Consider the poset (P(S), ⊂) and its associated scwol Y S .Then Y S = Π v∈S Y {v} .Moreover for any v ∈ S the scwol Y {v} , also denoted Y v , has two vertices ∅ and {v} and a single edge e v with i(e v ) = ∅ and t(e v ) = {v}.
Example 3.5.Consider a finite set S. For v ∈ S let Z {v} = Z v be the scwol consisting of two vertices ∅ and {v} and of two edges denoted (∅, {v}, ∅) and (∅, {v}, {v}) and the edges can be described as , where i(A, B, λ) = A and t(A, B, λ) = B.This example seems artificial at this point but will be quite useful later as the geometric realization of Z S is a torus T |S| and its fundamental group is Z |S| .Indeed we are particularly interested in the case where S is the vertex set of a complete Dyer graph Γ for which all vertices are labeled by ∞.
A simple complex of groups G(X ) = (G v , ψ α ) over a scwol X is given by the following data: , with the following compatibility condition: ψ αβ = ψ α ψ β whenever defined.
A simple complex of groups G(X ) = (G v , ψ α ) over a scwol X is called inclusive if it additionally satisfies the following condition: for each α ∈ E(X ) we have G i(α) < G t(α) and ψ α (g) = g for all g ∈ G i(α) .We will only be considering simple inclusive complexes of groups.These are restrictions on the more general definition of complexes of groups which can be found in [BH99].
Definition 3.6.The product G(X ) × G(Y) of two simple complexes of groups G(X ) and G(Y) is the simple complex of groups over the scwol X × Y given by the following data: As G(X ) and G(Y) are simple complexes of groups so is G(X ) × G(Y).
Similarly to the definition of products of scwols this definition can be extended to finite products of simple inclusive complexes of groups.The product Π i∈[n] G(X i ) of simple complexes of groups G(X i ), i ∈ [n], is the simple complex of groups over the scwol Π i∈[n] X i given by the following data: We will now give fundamental examples of complexes of groups and products of complexes of groups over the scwols introduced in Examples 3.4 and 3.5.
Example 3.7.We consider the scwols defined in Example 3.4.For every v ∈ S we choose a positive integer p v .Let C v be the finite cyclic group of order p v .For v ∈ S, let D(Y v ) be a simple complex of groups over Y v by choosing G ∅ = e the trivial group, G {v} = C v and ψ ev the trivial map.We define a simple complex of groups D(Y S ) over Y S as follows: so there is a canonical inclusion ψ e : G A → G B .These inclusions satisfy the compatibility condition.
We indeed have D(Y S ) = Π v∈S D(Y {v} ).
Example 3.8.For a finite Coxeter system (W, S) we can define W(Y S ) a simple complex of groups over Y S as follows: 1. for A ∈ V (Y S ) we choose the local group to be W A , the subgroup of W generated by A, 2. for e ∈ E(Y S ) with i(e) = A and t(e) = B we have A ⊂ B so there is a canonical inclusion ψ e : W A → W B .These inclusions satisfy the compatibility condition.
In general in this case we have Example 3.9.We consider the scwols defined in Example 3.5.We can always define a trivial complex of groups over a scwol.The product of trivial complexes of groups will again be trivial.This leads to the following notation.For every v ∈ S we define a simple complex of groups D(Z v ) over each scwol Z v by choosing G ∅ = G {v} = e the trivial group and ψ (∅,{v},∅) = ψ (∅,{v},{v}) the trivial map.Similarly we define a simple complex of groups D(Z S ) by choosing G A = e the trivial group for every A ∈ V (Z S ) and ψ (A,B,λ) the trivial map for every (A, B, λ) ∈ E(Z S ).We have Assume the scwol X is connected, i.e. there is only one equivalence class on V (X ) for the equivalence relation generated by (i(α) ∼ t(α) for every edge α ∈ E(X )).One can define the fundamental group of a complex of groups G(X ) over a scwol X .For simplicity and as it suffices for the cases we consider, we give the following characterization.Definition 3.10.Consider a simple complex of groups G(X ) over a connected scwol X .Assume each group G v is finitely presented with subject to the relations: Different choices of T will give isomorphic fundamental groups.So we can consider π 1 (G(X )) = π 1 (G(X ), T ) for any choice of maximal tree T .Moreover the subgroup of π 1 (G(X ), T ) generated by {α + , α ∈ E(X )} corresponds to the fundamental group of the scwol X .Lemma 3.11.For two simple inclusive complexes of groups G(X ) and G(Y) we have π Proof.We start by choosing maximal trees ) and consider the tree T X × Y spanned by the edges Since T X and T Y are maximal trees, so is T X × Y .To prove the statement we give explicit homomorphisms φ : π ) and show that their composition is the identity.Recall that the generating set of π This generating set is subject to the relations in π 1 (G(X )) and π 1 (G(Y)) and to the commutation relations {st There is a lot of redundancy in the generators of the group π 1 (G(X ) × G(Y), T X × Y ).In particular: 2. The previous statement implies that for every γ ∈ T X and every v ∈ V (Y) we have (γ, v) + = e.So we can do a similar construction as for the previous statement to show that for all we have on one hand (α, β) + = (α, t(β)) + (i(α), β) + = (α, w) + (v, β) + and on the other hand we have (α, β) . By the two previous statements this implies that for v ′ ∈ V (X ) and We define the map φ : , which respects the composition and sends an edge in ) .Similar statements hold when choosing v ∈ V (Y) and α ∈ E(Y).So the relations in π 1 (G(X )) and π 1 (G(Y)) are respected under φ.By the consequences listed above the commutation relations are also satisfied.Indeed for α ∈ E(X ), by using statement 7.There is a corresponding equality for α ∈ E(Y), w ∈ V (X ), t ∈ S w .So φ is a homomorphism.
We define the map ξ : Example 3.12.In Example 3.7 the fundamental group of D(Y S ) is × v∈S C v .In Example 3.9 the fundamental group of D(Z S ) is Z |S| .In Example 3.8 the fundamental group of W(Y S ) is the Coxeter group W .
We will now consider morphisms.Consider two scwols X and Y.A morphism of scwols f : X → Y is a map that sends V (X ) to V (Y), sends E(X ) to E(Y) and such that: where Ad(φ(α)) is the conjugation by φ(α).
There is always a morphism from the complex of groups to the fundamental group of the complex of groups.
Example 3.13.Consider the complex of groups D(Y S ) given in Example 3.7.Its fundamental group is Definition 3.16.A complex of groups G(X ) over a scwol X is developable if there exists a morphism φ from G(X ) to some group G which is injective on the local groups.
Remark 3.17.This definition is not the original definition given in III.C.2.11 [BH99] but it is equivalent to it by Corollary III.C.2.15 in [BH99] and better suited to our use.
Let G(X ) be a complex of groups over a scwol X , let G be a group and φ : G(X ) → G a morphism.The development of X with respect to φ is the scwol C(X , φ) given as follows: 1. its vertices are where α, β are composable and Note that by Theorem III.C.2.13 in [BH99], C(X , φ) is indeed well-defined.Moreover there is an action of G on C(X , φ) where G\ C(X , φ) = X .
As for simplicial complexes we can define geometric realizations of scwols.For a scwol X denote its geometric realization by | X |.If a scwol does not have multiple edges, this construction coincides with the geometric realization of simplicial complexes.This is the only case we will need in this article and details on the general construction can be found in Chapter III.C.1 in [BH99] and edges where i(g, (∅, {v}, ∅)) = (g, ∅), and t(g, (∅, {v}, ∅)) = (g, {v}), and i(g, (∅, {v}, {v})) = (g, ∅), and t(g, (∅, {v}, {v})) = (gx −1 v , {v}).So it is a line, each vertex has either two incoming or two outgoing edges.The group Z = x v acts by translation.There are two orbits, one corresponding to the vertices with incoming edges, one to the vertices with outgoing edges.A geometric realization of C(Z {v} , φ {v} ) is the real line R, where we identify (e, ∅) ∈ C(Z {v} , φ {v} ) with 0 ∈ R, and (e, {v}) with 0.5 and (x −1 v , {v}) with −0.5.Since we want Z to act by isometries, this means that for every x k v ∈ Z we identify the vertex (x k v , ∅) with k ∈ R and (x k v , {v}) with k + 0.5 ∈ R. Using the product structure we get that R |S| with the Euclidean metric is a geometric realization of C(Z S , φ S ) on which Z |S| acts by translation.

Coxeter groups and the Davis-Moussong complex
The discussion of the Davis-Moussong complex is based primarily on [Dav08].We will omit most proofs as they can be found in the literature, in particular in [Dav08] and [Bou02].
Let S be a finite set.Let M = (m(s, t)) s,t∈S be a symmetric matrix with m(s, t) ∈ N ∪{∞}, m(s, s) = 1 and m(s, t) = m(t, s) ≥ 2 if s = t.Such a matrix is called a Coxeter matrix.The Coxeter group associated to M is given by the following presentation W = s ∈ S | (st) m(s,t) = 1 for all s, t ∈ S , where m(s, t) = ∞ means there is no relation given between s and t.The pair (W, S) is called a Coxeter system.Consider the S × S matrix c defined by c st = cos(π − π/m(s, t)), the matrix c is called the cosine matrix of the Coxeter matrix M. When m(s, t) = ∞, we intepret π/∞ to be 0 and cos(π − π/∞) = −1.
The following fact states a classical result giving a necessary and sufficient condition for a Coxeter group to be finite.Fact 3.21 (Theorem 6.12.9 [Dav08]).A Coxeter group W is finite if and only if the cosine matrix c is positive definite.
For T ⊂ S let W T be the subgroup of W generated by T .Consider the poset of spherical subsets S = {T ⊂ S | W T is finite } ordered by inclusion.In an abuse of notation, let us also write S for the scwol associated to the poset S. Let W(S) be the complex of groups over S where the local group at T ∈ S is W T and for an edge (R, T ) the associated map ψ (R,T ) : W R → W T is the inclusion ψ (R,T ) (r) = r for every r ∈ R. The fundamental group of W(S) is W and there is an injective morphism φ = (φ T , φ((R, T ))) where φ T : W T → W is the inclusion and φ((R, T )) = e for every edge (R, T ).Let C(S, φ) be the development of W(S) with respect to φ.Let us also consider the poset W S = T ∈S W/W T , called the poset of spherical cosets.In a similar abuse of notation, let us also write W S for the scwol associated to the poset W S.
where (wW R , (R, T )) is an edge from the vertex (wW R , R) to the vertex (wW T , T ).In particular there is an edge from a vertex (wW R , R) to a vertex (w ′ W T , T ) if and only if R ⊂ T and wW T = w ′ W T (i.e.w ′−1 w ∈ W T ).
Proof.It follows from Remark 3.22 that the two scwols have the same set of vertices.For the edges note that for wW R , w ′ W T ∈ W S, we have wW R ⊂ w ′ W T if and only if R ⊂ T and w ′−1 w ∈ W T .So using Remark 3.22, the sets of edges also coincide.
Coxeter polytope From now on, assume W is finite.Let us recall the canonical representation of W . Consider Π = {α s | s ∈ S} and V = ⊕ s∈S R α s .Let •, • : V × V → R be the scalar product on V given by α s , α t = − cos( π m(s,t) ).The canonical representation of W on GL(V ) is given by ρ : W → GL(V ) with ρ(s)(x) = x − 2 α s , x α s , for s ∈ S, x ∈ V .The scalar product on V is ρ(W )invariant.There is a dual basis Π * = {α * s | s ∈ S}, satisfying α * s , α t = 0 if s = t and α * s , α s = 1.We choose x 0 = s∈S α * s ∈ V .The Coxeter polytope of (W, S), denoted Cox(W ), is the convex hull of {ρ(w)(x 0 ) ∈ V | w ∈ W }. It is endowed with the Euclidean metric.Note that its interior is non-empty.For a subset T ⊂ S, let Π T = {α s | s ∈ T } and V T be the subvector space of V spaned by T and t u : V → V be the translation by the vector u.This translation sends ρ(w)(x 0,T ) to ρ(w)(x 0 ) for every w ∈ W T .Specifically it is an isometry from Cox(W T ) to Cox T (W ).The General Case We now consider any Coxeter group W , so W need not necessarily be finite.We put a coarser cell structure on W S (or equivalently on C(S, φ)) to build the Davis-Moussong complex Σ by identifying each subposet (W S) ≤wW T ,T ∈S , which is isomorphic to the poset W T (S ≤T ), with a Coxeter polytope Cox(W T ).So we can give the following description of Σ.
Theorem 3.25 ([Dav08] Proposition 7.3.4.).There is a natural cell structure on Σ so that 1. its vertex set is W , its 1-skeleton is the Cayley graph Cay(W, S), and its 2-skeleton is a Cayley 2-complex, 2. each cell is a Coxeter polytope, 3. the link Lk(v, Σ) of each vertex is isomorphic to the abstract simplicial complex S >∅ , 4. a subset of W is the vertex set of a cell if and only if it is a spherical coset, 5. the poset of cells in Σ is W S.
Note that the Cayley graph Cay(W, S) is considered to be undirected, hence there are no double edges between vertices, even though all elements of S have order 2 in W . Furthermore all edges in Cay(W, S) are labeled, hence the edges of Σ are labeled.This labeling coincides with the labeling of vertices in Lk(v, Σ).Now that we have an appropriate description of Σ, let us state the following geometric property of Σ.
A simplicial complex L with piecewise spherical structure has simplices of size ≥ π/2 if each of its edges has length ≥ π/2.Such a simplicial complex is a metric flag complex if the following condition holds: Suppose {v 0 , . . ., v k } is a set of pairwise adjacent vertices of L. Put c ij = cos(d(v i , v j )).Then {v 0 , . . ., v k } spans a simplex if and only if the matrix (c ij ) is positive definite.Then Moussong's Theorem is the consequence of the following lemmata.
Lemma 3.27 ([Dav08] Lemma 12.3.1.).Let Lk be the link of a vertex in Σ with its natural piecewise spherical structure inherited from Σ. Then Lk is a simplicial complex and has simplices of size ≥ π/2.Moreover, it is a metric flag complex.
Note that using Fact 3.21 the set of vertices of Lk is S and T ⊂ S spans a simplex if and only if W T is finite.Moreover the distance between two vertices in Lk is given by d(v, w) = π − π/m(v, w).
Lemma 3.28 (Moussong's Lemma [Dav08] Lemma I.7.4.).Suppose L is piecewise spherical simplicial cell complex in which all cells are simplices of size ≥ π/2.Then L is CAT(1) if and only if it is a metric flag complex.

Right-angled Artin groups and the Salvetti complex
Every Coxeter group has an associated Artin group.We will concentrate on the class of right-angled Artin groups and present their analog to the Davis-Moussong complex, the Salvetti complex S Γ .An extensive discussion of right-angled Artin groups can be found in Charney's survey [Cha07].
Given a simplicial graph Γ, with vertex set V and edge set E, the associated right-angled Artin group A(Γ) is given by the following presentation If Γ has no edges A(Γ) is the free group of rank |V |, if Γ is a complete graph A(Γ) is the free abelian group of rank |V |.
Salvetti complex S Γ Let Γ be a simplicial graph with vertex set V .For any set of pairwise adjacent vertices V ′ = {v 1 , . . ., v n } consider the corresponding generators x i = x v i and let for every i ∈ {1, . . ., n}}.
Note that for two sets V ′ , V ′′ ⊂ V of pairwise adjacent vertices we have V ′ = V ′′ if and only if C(V ′ ) = C(V ′′ ).The Salvetti complex S Γ is a cube complex with vertex set A Γ and the cubes are sets of vertices of the form aC(V ′ ) for some a ∈ A(Γ) and V ′ ⊂ V a set of pairwise adjacent vertices of Γ.The Salvetti complex S Γ is known to be a CAT(0) cube complex by [CD95].
4 The cell complex Σ The goal of this section is to show geometrically that Dyer groups are CAT(0) by constructing an appropriate Euclidean cell complex Σ.The first step is to construct a scwol C associated to a Dyer group.The scwol C encodes the necessary information to build Σ.The vertices of C will correspond to subcomplexes of Σ and the edges of C will encode identifications between subcomplexes of Σ. Finally we will also be able to interpret C as a simplicial subdivision of the complex Σ.We will first focus on spherical Dyer groups D which factor as a direct product of a finite Coxeter group and cyclic groups.We start with the construction of a scwol X associated to a spherical Dyer group D and then define a complex of groups D(X ).The scwol C will be the development of the complex of groups D(X ).The second subsection will discuss this for general Dyer groups.The third subsection will be devoted to the Euclidean cell complex Σ.We extend the map m : Proof.Assume D is a spherical Dyer group.Then the restriction of c to V 2 × V 2 is positive definite.Since additionally Γ is a complete Dyer graph, this implies that the matrix c is positive definite.Now assume the cosine matrix c associated to (Γ, f, m) is positive definite.Consider the matrix M = (m(u, v)) u,v∈V .Then the cosine matrix c associated to (Γ, f, m) is equal to the cosine matrix of the Coxeter matrix M as defined in Section 3.2.So by Fact 3.21 the cosine matrix c is positive definite if and only if the Coxeter group associated to M is finite.So we have m(u, v) = ∞ for all u, v ∈ V .Moreover since Γ is a Dyer graph this also implies that the restriction of c to V 2 × V 2 is positive definite.So the graph Γ is complete and D 2 is a finite Coxeter group by Fact 3.21.Hence D is a spherical Dyer group.

A combinatorial structure for spherical Dyer groups
Let X = X (Γ) be the scwol with set of vertices We call X the scwol associated to the spherical Dyer graph Γ. Similarly to the group D we can also describe X as a direct product of scwols.
For the edges note that (X, Y, ω) ∈ E(X ) if and only if X i ⊂ Y i for every i ∈ {2, p, ∞} and at least one of those inclusions is strict and ω ⊂ Y ∞ \ X ∞ .
We define the simple complex of groups D(X ) over the scwol X .For each These maps are all injective.Note that they do not depend on ω.We also introduce the morphism only depends on ω so we will write φ(X, Y, ω) = φ(ω) = Π v∈ω x v .Also note that each local group D f X is finite.and the outgoing star is defined similarly Moreover the product decomposition of C induces a product decomposition of the incoming star and as such also a product decomposition for every St in (gY, C).Moreover for a vertex hZ ∈ St in (gY, C), the star St in (hZ, C) is a subscwol of St in (gY, C).

A combinatorial structure for general Dyer groups
Let us now give a similar construction with analogous results for general Dyer groups.Let (Γ, f, m) be a Dyer graph and D = D(Γ) be the associated Dyer group.We note V = V (Γ).Let X = X (Γ) be the scwol with set of vertices V (X ) = {X ⊂ V | D(Γ X ) is a spherical Dyer group} and set of edges The main difference with the spherical case, is the set of vertices of X .Indeed we do not consider all subsets X ⊂ V but only those for which Γ X is complete and the group D f X = D X 2 ∪Xp is finite.We also define a complex of groups D(X ) over X .For each X ∈ V (X ) let the local group be D f X = D X 2 ∪Xp and for each edge (X, Y, ω) ∈ E(X ), let ψ (X,Y,ω) : D f X → D f Y be the natural inclusion.By [Dye90], these maps are all injective.The local groups are all finite.We also introduce the morphism φ : D(X ) → D where φ X : D f X → D is the natural inclusion and φ(X, Y, ω) = φ(ω) = Π v∈ω x v (this element is well defined since ω ⊂ V ∞ and Γ ω is complete).As in the spherical case we can write D(X (Γ)) and φ Γ when also considering the same construction on a subgraph.As before we are interested in the development of the complex of groups D(X ) so we first show that D(X ) is developable.
Example 4.9.Consider the Dyer graph Γ m,p , given again in Figure 4, and the Dyer group D m,p from Example 2.6.The associated scwol X m,p is drawn in Figure 5.Its vertex set is V (X m,p ) = {∅, {a}, {b}, {c}, {d}, {a, b}, {b, c}, {c, d}}.Lemma 4.10.The scwol X is isomorphic to the union of scwols Y = Y ∈V (X ) X Y , where X Y is the scwol associated to the spherical Dyer group D Y .The fundamental group of D(X ) is D. In particular, the complex of groups D(X ) is developable.
Proof.First we compare the sets of vertices.If Y ∈ V (X ) then Y ∈ V (X Y ) so Y ∈ V (Y).On the other hand if Y ∈ V (Y), we have Y ∈ V (X Z ) for some Z ∈ V (X ) so Y ⊂ Z so D Y is spherical.This implies V (X ) = V (Y).Now we compare the sets of edges.If e = (X, Y, ω) ∈ E(X ) then e ∈ E(X Y ) so e ∈ E(Y).On the other hand if e ∈ V (Y), e ∈ V (X Z ) for some Z ∈ V (X ) so e = (X, Y, ω) with X Y ⊂ Z and ω ⊂ (Y \X) ∞ so e ∈ E(X ).This implies E(X ) = E(Y).We can now apply Seifert-van Kampen to Y.The set V (X ) is finite and each scwol X Y is connected.We have ∅ ∈ V (X Y ) for all Y ∈ V (X ) and ∅ is adjacent to any vertex in any X Y .So Y ∈V (X ) X Y is nonempty and connected.We can then use the presentations to see that the fundamental group of D(X ) is D. Finally, by [Dye90], the maps φ X : D f X → D are all injective.So D(X ) is developable.Remark 4.11.Since the complex D(X ) is developable we can describe its development C = C(X , φ).Since D f X < D and the maps φ X are canonical inclusions, we will identify the image φ where associated to vertices, then build Σ and finally show that Σ is CAT(0) using Moussong's Lemma 3.28.Let (Γ, f, m) be a Dyer graph, D = D(Γ, f, m) the associated Dyer group, X = X (Γ) the associated scwol and D(X ) the associated complex of groups.Consider the injective morphism φ : D(X ) → D given by the natural inclusion φ X : D f X → D and φ(X, Y, ω) = φ(ω) = Π v∈ω x v .As in the previous section we construct the development C = C(X , φ).

Elementary building blocks
with its standard cubical structure.Its set of vertices is P(Y ), where 0 ∈ R |Y | corresponds to ∅ ∈ P(Y ).For v ∈ V p let Stern(v) be the f (v)-branched star where each edge of the star is identified with [0, 1].Its center is denoted c v and its tips are identified with the elements of C f (v) the finite cyclic group of order f (v).For Y ∈ V (X ) with Y = Y p let Stern(Y ) be the product of stars Π v∈Y Stern(v) endowed with the ℓ 2 metric.So its vertex set is Π The cell complex Cc(Y ) To every Y ∈ V (X ) we associate a Euclidean cell complex Cc(Y ) as follows.Let Cc(Y ) be the product Cox(Y 2 )×[0, 1] |Y∞| ×Stern(Y p ) endowed with the ℓ 2 metric.Each of its factors is piecewise Euclidean, so it is a piecewise Euclidean cell complex.In particular Cc  Figure 6: The cell complexes associated to some vertices of X m,p .
The cell complex Σ(gY ) Let gY ∈ V (C).We now describe the subcomplexes of Σ associated to vertices of C. We start by identifying the vertex set of Cc(Y ) with a subset of V (St in (Y, C)) and more generally with a subset of V (St in (gY, C)).Let V p (gY ) be the following subset of V (St in (gY, C)): Lemma 4.15.The map j : V (Cc(Y )) → V p (e Y ) given by j(w, λ, hZ) = wφ(λ)hZ is bijective.Moreover it induces a bijective map j g : V (Cc(Y )) → V p (gY ) with j g (w, λ, hZ) = g • j(w, λ, hZ) = gwφ(λ)hZ.
Proof.For every vertex kZ ∈ V p (e Y ), we have Z ⊂ Y p and there is a unique So the map j is bijective.Finally kZ ∈ V p (gY ) if and only if g −1 kZ ∈ V p (e Y ).So the map j g is bijective.
For gY ∈ C let Σ(gY ) be the piecewise Euclidean cell complex isometric to Cc(Y ) with vertex set V (Σ(gY )) = V p (gY ) induced by the map j g .Note that if hY = gY the map j g • j −1 h induces an isometry of Σ(gY ) so Σ(gY ) is well defined.More precisely U ⊂ V p (gY ) is the vertex set of a cell of Σ(gY ) if and only if j −1 g (U) is the vertex set of a cell in Cc(Y ).
We now discuss identifications of subcomplexes.Indeed for gY ∈ V (C) and hZ ∈ St in (gY, C), we have V p (hZ) ⊂ V p (gY ).The following lemma shows that this inclusion induces an isometric embedding of the cell complex Σ(hZ) into Σ(gY ).
) is an isometric embedding preserving the cell structure.Indeed this holds for the restrictions to Cc(Z 2 ) (where ι(w) = h 2 w which corresponds to identifying Cox( Then the map ι : Σ(hZ) → Σ(e Y ) is induced by j −1 h • ι h • j e so ι is an isometric embedding preserving the cell structure.
The next lemma discusses how St in (gY, C) can be interpreted as a simplicial subdivision of Σ(gY ).
Proof.Let Y ∈ V (X ) and g ∈ D. Then using Lemma 3  where we identify Σ(hZ) with ι(Σ(hZ)) ⊂ Σ(gY ) whenever hZ ∈ St in (gY, C).So by Lemma 4.16, Σ has a well-defined piecewise Euclidean metric.We endow Σ with the associated length metric.The set of vertices of Σ is The action of D on V p (C) induces an action by isometries of D on Σ, in particular for d ∈ D we have d • Σ(gY ) = Σ(dgY ).
Lemma 4.19.The scwol C describes a simplicial subdivision of Σ.In particular this implies that Σ is a simply connected metric space.
Proof.Since C = gY ∈V (C) St in (gY, C) and by Lemma 4.17 every St in (gY, C) is a simplicial subdivision of Σ(gY ) preserved by ι, the complex C is a simplicial subdivion of Σ.This induces a metric on C with respect to which the geometric realization | C | is isometric to Σ.So Σ is a well-defined simply connected metric space.
We are finally in a position to show that Σ is CAT(0).Since Σ is simply connected, we only need to understand its local structure, so we are back to studying links of vertices.In order to have a precise description of the links of vertices we introduce an edge labeling of Σ by V (Γ).
Edge labeling Let hY ∈ C. We start by labeling the edges of Σ(hY ) by elements of Y .To define this edge labeling we identify when two vertices of Σ(hY ) are adjacent and then give the corresponding label.Let kX, lZ ∈ V p (hY ), i.e. they are vertices of Σ(hY ).Then kX and lZ are adjacent in Σ(hY ) if and only if their pre-images j −1 h (kX), j −1 h (lZ) ∈ V (Cc(Y )) are adjacent in Cc(Y ).This leads to the following characterization and labeling of edges by Y ⊂ V (Γ).The vertices kX, lZ ∈ V p (hY ) are adjacent in Σ(hY ) if and only if one of the following holds 1. X = Z and k −1 l = x v for some v ∈ Y 2 .In this case we label the edge by v ∈ Y 2 ⊂ V (Γ).
2. X = Z and k −1 l = x ±1 v for some v ∈ Y ∞ .In this case we label the edge by v ∈ Y ∞ ⊂ V (Γ).
3. X ⊂ Z and Z \ X = {x v } for some v ∈ Y p and k −1 l ∈ x v .In this case we label the edge by v ∈ Y p ⊂ V (Γ).
4. Z ⊂ X and X \ Z = {x v } for some v ∈ Y p and k −1 l ∈ x v .In this case we label the edge by v ∈ Y p ⊂ V (Γ).
Note that for h ′ Y ′ ∈ St in (gY, C), the labeling of an edge in Σ(h ′ Y ′ ) is invariant under the inclusion ι : Σ(h ′ Y ′ ) → Σ(gY ).Moreover the labeling of edges in Σ(e Y ) is invariant under the action of D f Y .So this defines a labeling by V (Γ) of the edges of Σ.Note that this edge labeling is invariant under the action of D.
Remark 4.20 (Links of vertices).As our goal is to apply Moussong's Lemma to Σ we need to understand links of vertices in Σ.We start with links of vertices in Σ(gY ).This is crucial to prove later that Σ is CAT(0).Let hY ∈ V (C), kX ∈ V p (hY ).The edge labeling on Σ and Σ(gY ) induce a vertex labeling l : V (Lk(kX, Σ)) → V , which restricts to l : V (Lk(kX, Σ(hY ))) → Y .Using the map j h in Lemma 4.15 and Remark 4.13, we see that Lk(kX, Σ(hY )) is the flag complex over the join Γ Y 2 ⋆ Γ Y∞ ⋆ Γ Yp\X ⋆ (⋆ v∈X {v i | 1 ≤ i ≤ f (v)}).The vertex labeling is given by l(v) = v for every v ∈ Y 2 ∪ Y ∞ ∪ Y p \ X and l(v i ) = v for every v i ∈ {v i | v ∈ X, 1 ≤ i ≤ f (v)}.By Remark 4.13, the edge length in Lk(kX, Σ(hY )) is given by d(v, w) = π − π/m(l(v), l(w)).In particular Lemma 4.1 implies that Lk(kX, Σ(hY )) is a metric flag complex.It follows that v, w are adjacent vertices in Lk(kX, Σ(hY )) if and only if l(v) = l(w).As this holds for every gY ∈ V (C), it implies that if v, w are adjacent vertices in Lk(kX, Σ), we have l(v) = l(w).So for pairwise adjacent vertices v 1 , . . ., v n ∈ Lk(kX, Σ), we have l(v i ) = l(v j ) for every i = j.So in a slight abuse of notation we will write v i = l(v i ) ∈ V when considering pairwise adjacent vertices v 1 , . . ., v n ∈ Lk(kX, Σ).
Let us now give some more details on the link of vertices in Σ. (iv) For v ∈ Y , the vertex v ∈ Lk(Y, Σ) corresponds to an edge between Y and x t v (Y \ {v}), for some 1 ≤ t ≤ f (v).In this case fix g v = e.
(iv) If v ∈ Y , we have h v Z v = x t v (Y \ {v}) for some 1 ≤ t ≤ f (v) and we set As Z = Y ∪ {v 1 , . . ., v k } ∈ V (X ), we have gZ = h v φ(λ v ) −1 Z and Z v ⊂ Z, so h v Z v ∈ V (St in (gZ, C)).As additionally Z v ⊂ Z ∩ V p , we have h v Z v ∈ V p (gZ).
We now have the necessary tools to show the following statement.
Theorem 4.22.The cell complex Σ is CAT(0).Proof.By [BH99] Theorem II.5.4,Σ is CAT(0) if and only if it is simply connected and the link of every vertex is CAT (1).By Lemma 4.19, the cell complex Σ is simply connected.Let us now prove that the link of every vertex is CAT(1) by using Moussong's Lemma 3.
x u , x v ] m(e) for all e = {u, v} ∈ E , where [a, b] k = aba . . .k for any a, b ∈ D, k ∈ N and we denote the identity with e.

Figure 1 :
Figure 1: Dyer graph Γ m,p for some m, p ∈ N ≥2 x u , x v ] m(e) for all e = {u, v} ∈ E , where [a, b] k = aba . . .k for any a, b ∈ W , k ∈ N and we denote the identity with e.Note that for an edge e = {u, v} ∈ E the relation

Figure 3 :
Figure 3: The graph Ω m,p build out of the Dyer graph Γ m,p for some m, p ∈ N ≥2 .There are two types of vertices: V ⊂ V (Λ m,p ) and {v ′ | v ∈ V ∞ }.Every vertex and every edge is labeled by 2 if not specified otherwise.
and for α ∈ E(Y S ) let φ S (α) = e ∈ × v∈S C v .The morphism φ S = (φ S A , φ S (α)) : D(Y S ) → × v∈S C v is injective on the local groups.Example 3.14.Consider the complex of groups W(Y S ) given in Example 3.8.Its fundamental group is π 1 (W(Y S )) = W .For A ∈ V (Y S ), let φ A : W A → W with φ(g) = g and for α ∈ E(Y S ) let φ(α) = e ∈ W .The morphism φ = (φ A , φ(α)) : W(Y S ) → W is injective on the local groups.Example 3.15.Consider the complex of groups D(Z S ) given in Example 3.9.Its fundamental group is π 1 (D(Z S )) = Z |S| .For the notation: let e be the trivial element in Z |S| and let x s , s ∈ S be the standard generators of Z |S| .For A ∈ V (Z S ), let φ S A : e → Z |S| with φ S (e) = e and for (A, B, λ) ∈ E(Z S ) let φ S ((A, B, λ)) = Π s∈λ x s .The morphism φ S = (φ S A , φ S (α)) : D(Z S ) → Z |S| is injective on the local groups.
. The action of G on C(X , φ) induces an action of G on | C(X , φ)|.If we require the action of G on | C(X , φ)| to be by isometries, putting a metric on | C(X , φ)| corresponds to putting a metric on | X | as G\| C(X , φ)| = | X |.Example 3.18.Consider the complex D(Y S ) and the morphism φ S : D(Y S ) → Π v∈S C v from Example 3.13.One can check that the development ) and the edges are oriented from the tips to the center.The group C v acts by rotation and stabilizes the central vertex.For each v ∈ S choose l v > 0. Let Stern(v) be the geometric realization of C(Y v , φ {v} ) as follows: for g ∈ C v consider the interval I g = [0, l v ] then Stern(v) = g∈Cv I g / ∼ where 0 ∈ I g ∼ 0 ∈ I e .Note that C v acts by isometries on Stern(v).The space Stern(S) = Π v∈S Stern(v) with the product metric is a geometric realization of C(Y S , φ S ), due to the product structure of C(Y S , φ S ).Moreover Π v∈S C v acts by isometries on Stern(S).Example 3.19.Consider the complex W(Y S ) and the morphism φ : W(Y S ) → W from Example 3.14.The development C(Y S , φ) is a scwol with set of vertices {(gW A , A) | A ⊂ S, gW A ∈ W/W A } and {(gW A , (A, B)) | A B, gW A ∈ W/W A } as set of edges where i(gW A , (A, B)) = (gW A , A) and t(gW A , (A, B)) = (gW B , B).It is the scwol associated to the poset W P(S) = T ⊂S W/W T , the poset of parabolic cosets ordered by inclusion.In Section 3.2 we will introduce the Coxeter polytope C W of W , which is a geometric realization of C(Y S , φ).Example 3.20.Consider the complex D(Z S ) and the morphism φ S : D(Z S ) → Z |S| from Example 3.15.One can check that the development C(Z S , φ S ) is the product Remark 3.22.The set of vertices of C(S, φ) is {(wW T , T ) | T ∈ S, wW T ∈ W/W T }.The set of edges of C Lemma 3.24 (Lemma 7.3.3[Dav08]).The poset W S and the face poset F(Cox(W )) of Cox(W ) are isomorphic.Specifically the correspondance wW T → w Cox T (W ) induces an isomorphism of posets.So we can identify W S and hence C(S, φ) with the barycentric subdivision of the Coxeter polytope Cox(W ).Thus identifying | C(S, φ)| isometrically with Cox(W ).The metric on | C(S, φ)| induced by the identification with Cox(W ) is called the Moussong metric.In particular for wW T ∈ W S the geometric realisation |W S ≤wW T | ⊂ |W S | is identified with the face w Cox T (W ).

A
Dyer group D = D(Γ, f, m) is spherical if its underlying graph Γ is complete and the subgroup D 2 is a finite Coxeter group.If D = D(Γ, f, m), we also say that Γ = (Γ, f, m) is a spherical Dyer graph.In this section we will assume D is a spherical Dyer group.In particular we then have D = D 2 ×D p ×D ∞ , where D 2 is a finite Coxeter group, D p is a direct product of finite cyclic groups and D ∞ = Z |V∞| .As with Coxeter groups, we can characterize spherical Dyer groups through the cosine matrix.Let (Γ, f, m) be a Dyer graph and let V = V (Γ) and E = E(Γ).
vertex.Let gY ∈ V (C).The incoming link Lk in (gY, C) is the full subscwol of C spanned by the vertices {hZ | ∃e ∈ E(C) : t(e) = gY and i(e) = hZ}.Similarly the outgoing link Lk out (gY, C) is the full subscwol of C spanned by the vertices {hZ | ∃e ∈ E(C) : i(e) = gY and t(e) = hZ}.The incoming star is the subscwol spanned by gY and its incoming link so it is the combinatorial join St in (gY, C) = {gY } ⋆ Lk in (gY, C)

Lemma 4. 16 .
Let gY ∈ V (C) and hZ ∈ St in (gY, C).The map ι : Σ(hZ) → Σ(gY ) induced by ι(v) = v for every vertex v ∈ Σ(hZ) is an isometric embedding preserving the cell structure.In particular we can identify Σ(hZ) with ι(Σ(hZ)).Proof.Since hZ ∈ St in (gY, C) if and only if g −1 hZ ∈ St in (Y, C) it suffices to consider the case g = e.For hZ ∈ St in (e Y, C) we can write

Figure 7 :
Figure 7: The subcomplexes associated to some vertices of the development of X m,p and their simplicial subdivision.
28. Consider gY ∈ V (Σ) with g = e.So gY = e Y = Y .Claim 1.Every edge in the link Lk(Y, Σ) of Y in Σ has length ≥ π/2.Proof.Since the vertex Y ∈ Σ is contained in Σ(gZ) if and only if gZ ∈ St out (Y, C) we can describe Lk(Y, Σ) as the union gZ∈Stout(Y,C) Lk(Y, Σ(gZ)), where Lk(Y, Σ(gZ)) is the link of Y in the subcomplex Σ(gZ).By Remark 4.20, the length of edges in Lk(Y, Σ(gZ)) satisfy d(u, v) = π − π/m(l(u), l(v)) ≥ π/2 as m(l(u), l(v)) ≥ 2. So each edge in Lk(Y, Σ) has length ≥ π/2.Claim 2. The link Lk(Y, Σ) of Y in Σ is metrically flag.Proof.Consider a set of pairwise adjacent vertices v 1 , . . ., v k ∈ Lk(Y, Σ).As mentioned in Remark 4.20, so l(v i ) = l(v j ), so we writev i = l(v i ) ∈ V .In particular Y ∪ {v 1 , . . ., v k } spans a complete subgraph of Γ.So v 1 , . . ., v k spana simplex in Lk(Y, Σ) if and only v 1 , . . ., v k span a simplex in Lk(Y, Σ(gZ)) for some gZ ∈ V (C).By Remark 4.20 Lk(Y, Σ(gZ)) is flag.So v 1 , . . ., v k ∈ Lk(Y, Σ) span a simplex in Lk(Y, Σ) if and only if there exists some gZ ∈ V (C) with Y ∈ Σ(gZ) and v 1 , . . ., v k ∈ Lk(Y, Σ(gZ)).By Lemma 4.21, this is the case if and only if Dyer groups are finite index subgroups of Coxeter groups The aim is now to show that every Dyer group is a finite index subgroup of a Coxeter group.From a given Dyer graph (Γ, f, m) we build a graph Λ with edge labeling m ′ .We then show that D(Γ, f, m) is a finite index subgroup of W (Λ). See Example 2.11 for a simple case.We define the undirected labeled simplicial graph Λ.Its set of vertices is V Coxeter group W ′ associated to the Dyer group D ′ such that W ′ = D ′ ⋊ (Z /2 Z) |Vp| .Question: do W and W ′ relate in any (meaningful) way?What can we say about their Davis-Moussong complexes?How do D and D ′ relate to each other?What are all the Dyer subgroups of a given Coxeter group?
C),we have that St in (Y, C) describes a simplicial subdivision of Cc(Y ).Hence St in (gY, C) is the scwol of this simplicial subdivision of Σ(gY ).This induces a metric on the geometric realization | St in (gY, C)| such that it is isometric to Σ(gY ).For the second statement note that ι is an isometric embedding, preserves the cell structure, and decomposes as a product.Additionally the canonical inclusion St in (hZ, C) → St in (gY, C) also decomposes as a product.The simplicial subdivision is preserved for each factor of the product decomposition.So the simplicial subdivision is preserved by ι.Example 4.18.Let m, p ∈ N ≥2 .We go back to the example of the Dyer graph Γ m,p with associated Dyer group D m,p and scwol X m,p given in Figure 1, Example 2.6 and Figure 5.