Commensurability of graph products

Tadeusz Januszkiewicz^{1} and Jacek Świa̧tkowski^{2}

Graph products are useful and pretty generalizations of both products and free products, intimately linked with right angled buildings. Part of their appeal is the generality: they can be studied in any category with products and direct limits.

The question that motivated the present paper was ”when are the graph products of two families of groups commensurable”. The inspiration came from a special case considered in [DJ] and from a conversation with Marc Bourdon on linearity of certain lattices in automorphism groups of right angled buildings.

Here is an answer to the simplest version of this question. Recall first that two groups G, G^{∗} are commensurable if there is a group H, isomorphic to a subgroup of a finite index in both G and G^{∗}; they are strongly commensurable if H has the same index in both G and G^{∗}.

Theorem 1. Let Γ be a finite graph, (G_{v})_{v∈V}, (G_{v}^{∗})_{v∈V} be two families of groups indexed by the vertex set of Γ. Suppose that for every v ∈ V, G_{v} and G_{v}^{∗} are strongly commensurable with the common subgroup H_{v}. Then the graph products G = Π_{Γ}(G_{v})_{v∈V}, and G^{∗} = Π_{Γ}(G_{v}^{∗})_{v∈V} are strongly commensurable: they share a subgroup of index Π_{v}[G_{v} : H_{v}].

We will prove a slightly more general result on graph products of pairs of groups. The proof uses two complementary descriptions of right angled building on which a graph product acts. One of them allows an easy identification of the group acting as the graph product, the other allows to compare subgroups.

Theorem 1 and its stronger version formulated in Section 4 (Corollary 4.3) have several interesting special cases discussed in Section 5.

1. Graph products of pairs.

1.0. Graphs. A graph Γ on the vertex set V = V (Γ) is an antireflexive symmetric relation on V. Thus our graphs have no loops and there is at most one undirected edge between two vertices. Graphs considered in this paper are always finite.

A full subgraph Γ^{+} < Γ on vertices W ⊂ V is the restriction of the relation to W. A graph is complete if there is an edge between any two vertices. A map of graphs f : Γ → Γ^{∗} is an injection of sets of vertices with the property that

if there is an edge between v, w then there is an edge between f(v), f(w). Thus our maps of graphs are inclusions.

1.1. Graph products. Let Γ be a finite graph, with vertex set V. Suppose for each v ∈ V one is given a pair of groups A_{v} < G_{v}. For S a complete subgraph of Γ define

1 Instytut Matematyczny Uniwersytetu Wrocławskiego and IM PAN 2 Instytut Matematyczny Uniwersytetu Wrocławskiego

both authors were supported by a KBN grant 5 P03A 035 20 G_{S} = Π_{v∈S}G_{v} × Π_{v∈V \S}A_{v}. The family of groups G_{S} together with obvious inclusions on factors of products gives a direct system of groups, directed by the poset P of complete subgraphs in Γ, empty set and singletons included (G_{∅} = Π_{v}A_{v}; G_{{v}} = G_{v}×Π_{w∈V \{v}}A_{w}).

The direct limit of this system

G = lim (G_{S})_{S∈P} = Π_{Γ}(G_{v}, A_{v})

is called the graph product along Γ of the family of pairs (G_{v}, A_{v}). To keep notation simple we will denote it for most of the time by G. Note that for A_{v} = {e}, we obtain ordinary graph products.

1.2. Graph products are functorial. If g : Γ → Γ^{∗} is a map of graphs, and if there is a family of group homomorphisms ω_{v} : G_{v} → G_{g(v)}^{∗}, such that ω_{v}(A_{v}) < A_{g(v)}^{∗} then we have induced maps ω_{S} : G_{S} → G_{g(S)}^{∗} which clearly commute with the maps of direct systems, and consequently induce a homomorphism

ω : G → G^{∗}.

If g is a surjection on the vertices and ω_{v} are all surjections, so is the induced homomorphism ω. If g is an embedding onto a full subgraph and ω_{v} are injections, so is the induced homomorphism.

1.3. Remark. It follows from functoriality above, that if Γ is a full subgraph of Γ^{∗} then graph product of any family of pairs along Γ^{∗} contains as a subgroup the graph product of that family of pairs restricted to Γ. In particular, groups G_{S} inject into G. Thus we can (and will) consider G_{S} as subgroups of G.

1.4. Presentations. Graph products can be given in terms of generators and relations. Suppose that each group G_{v} is given by presentation ⟨S_{v}|R_{v}⟩ and that Σ_{v} is a set of generators for the subgroup A_{v} expressed in terms of generators in S_{v}. Then the graph product G = Π_{Γ}(G_{v}, A_{v}) is given by the presentation ⟨∪_{v∈V} S_{v}| ∪_{v∈V} R_{v} ∪ C⟩, where C consists of commutators {sts^{−1}t^{−1}} whenever s ∈ S_{v}, t ∈ S_{w} and there is an edge between v and w in Γ, or whenever s ∈ S_{v}, t ∈ Σ_{w} for some v ≠ w.

1.5. Examples.

1. Graph product of pairs Π_{Γ}(G_{v}, A_{v}) along a complete graph Γ is the (direct) product Π_{v∈V} G_{v}.

2. If Γ is an empty graph (i.e. an empty relation on the vertex set V) then the graph product Π_{Γ}(G_{v}, A_{v}) is the free product of groups G_{{v}} = G_{v} × Π_{w∈V \{v}}A_{w} amalgamated along their common subgroup G_{∅} = Π_{v∈V} A_{v}.

3. Graph products (i.e. graph products with trivial subgroups A_{v}) of infinite cyclic groups are called right angled Artin groups.

4. Graph products of cyclic groups of order 2 are called right-angled Coxeter groups (i.e. Coxeter groups with exponents 2 or ∞ only).

2. The complex D_{G}.

2.1. Description of D_{G}. Let P be the realization of the poset P of complete subgraphs in Γ, i.e. the simplicial complex with the vertex set P and with simplices corresponding to flags (i.e. linearly ordered subsets) of P. For each S ∈ P let P_{S} be the subcomplex of P spanned by those vertices S′ ∈ P which contain S. Note that the poset of subcomplexes P_{S}, with the reverse inclusion is isomorphic to the poset P. Define a simplicial complex D_{G} = G × P/ ∼, where ∼ is the equivalence relation given by

(g_{1}, x_{1}) ∼ (g_{2}, x_{2}) iff for some S ∈ P we have x_{1} = x_{2} ∈ P_{S} and g_{1}^{−1}g_{2} ∈ G_{S} ⊂ G.

We shall denote the point in D_{G} corresponding to a pair (g, x) ∈ G × P by [g, x]. Group G acts on the complex D_{G} on the left by g · [g′, x] = [gg′, x].

2.2. Remark. The G action on D_{G} need not be effective. Its kernel is the product ΠN_{v} < ΠA_{v}, where N_{v} is the intersection of all G_{v} conjugates of A_{v}. Dividing by the kernel of the action is geometrically sound, and gives the reduced graph product of pairs. For example if all A_{v} are normal, the reduced graph product is just the graph product of quotients.

2.3. Complex of groups G(P). Denote by G(P) the simple complex of groups (in the sense of [BH], Chapter II.12) over the poset P defined by the directed system (G_{S})_{S∈P} of groups. In view of the injectivity discussed in 1.3, Theorems 12.18, 12.20 and Corollary 12.21 of [BH] imply

2.4. Proposition. The simplicial complex D_{G} is isomorphic to the development of the complex of groups G(P) corresponding to the family (i_{S})_{S∈P} of canonical inclusions i_{S} : G_{S} → G into the direct limit. In particular, D_{G} is connected and simply connected.

Moreover the the complex of groups associated to the action of G on D_{G} coincides with that of G(P).

2.5. D_{G} is a building. The complex D_{G} is well known and is sometimes called the right angled building associated to a graph product G, see [D] (Section 5) and [BH] (section 12.30 (2)). It is indeed a Tits building whose appartments are Davis complexes of the (right angled) Coxeter group which is the graph product of Z_{2}’s along Γ.

3. Another description of G and D_{G}.

3.1. Associated graph product along the complete graph. Given a finite graph Γ on the vertex set V and a graph product G = Π_{Γ}(G_{v}, A_{v}), denote by G^{c} the graph product of pairs (G_{v}, A_{v}) along the complete graph Γ^{c} on the vertex set V. Put ω^{c} : G → G^{c} to be the homomorphism given by functoriality as in 1.2, and note that ω^{c} is surjective.

Let P^{c} be the poset of complete subgraphs in Γ^{c} (including singletons and the empty graph), and let P^{c} be its realization. The inclusion Γ → Γ^{c} clearly induces an injective simplicial map p^{c} : P → P^{c} (where P is the realization of the corresponding poset for Γ).

3.2. Complex Δ_{G} and group G̃. Let D_{G^{c}} be the simplicial complex associated to the graph product G^{c} as in 2.1. Denote by π^{c} : D_{G^{c}} → P^{c} the simplicial map induced by the projection G^{c} × P^{c} → P^{c}. Put Δ_{G} := (π^{c})^{−1}(p^{c}(P)) and note that, since the action of G^{c} on D_{G^{c}} commutes with π^{c}, the subcomplex Δ_{G} ⊂ D_{G^{c}} is invariant under this action. Thus we will speak about the (restricted) action of G^{c} on Δ_{G}. Consider the universal cover Δ̃_{G} of Δ_{G}, with the action of the group G̃, which is the extension (induced by the covering Δ̃_{G} → Δ_{G}) of the group G^{c} by the fundamental group π_{1}(Δ_{G}).

3.3 Theorem. Groups G̃ and G are isomorphic, simplicial complexes D_{G} and Δ̃_{G} are equivariantly isomorphic, and the homomorphism G̃ → G^{c} induced by the covering Δ̃_{G} → Δ_{G} coincides with the map ω^{c} : G → G^{c}.

Proof: Let f : D_{G} → Δ_{G} ⊂ D_{G^{c}} be the map defined by f([g, x]) = [ω^{c}(g), p^{c}(x)]. This map is easily seen to be surjective and ω^{c}-equivariant. It induces then a morphism f_{∗} : G\\D_{G} → G^{c}\\Δ_{G} between the complexes of groups associated to the actions of G on D_{G} and of G^{c} on Δ_{G}.

Observe that for a vertex [g, S] ∈ D_{G} the isotropy subgroup of G at [g, S] describes as Stab(G, [g, S]) = g^{−1}G_{S}g. By substituting G with G^{c} in this observation we get easily that the homomorphism ω^{c} : G → G^{c} maps stabilizers in D_{G} isomorphically to stabilizers in D_{G^{c}}, and hence also in Δ_{G}. The morphism f_{∗} is then isomorphic on local groups. Since moreover the map between the underlying spaces (quotient spaces of the corresponding actions) associated to the morphism f_{∗} is a bijection, it follows that f_{∗} is an isomorphism of complexes of groups.

Let u : Δ̃_{G} → Δ_{G} be the universal covering map. As before, by natural equivariance, this map induces a morphism u_{∗} : G̃\\Δ̃_{G} → G^{c}\\Δ_{G} between the complexes of groups associated to the corresponding actions. It follows then from local injectivity of u that the stabilizers of G̃ in Δ̃_{G} are mapped isomorphically (by the homomorphism G̃ → G^{c} associated to the covering) to the stabilizers of G^{c} in Δ_{G}, hence u_{∗} is isomorphic on local groups. Combining this with equality of the underlying quotient complexes (which follows directly from the description of G̃), we get that u_{∗} is also an isomorphism of complexes of groups.

Now, since both complexes D_{G} and Δ̃_{G} are connected and simply connected, it follows that they are both equivariantly isomorphic with the universal covering of the complex of groups Π_{v∈V} G_{v}\\Δ_{G} acted upon by the fundamental group of this complex of groups. Thus the theorem follows.

3.4 Complex CX. Consider the family X = (X_{v})_{v∈V} of quotients X_{v} = G_{v}/A_{v}. Denote by C the poset consisting of all subsets Y in the disjoint union ∪X having at most one common element with each of the sets X_{v}. We assume that the empty set ∅ is also in C. Put CX to be the realization of the poset C, i.e. a simplicial complex with simplices corresponding to linearly ordered subsets of C. Alternatively, CX is the simplicial cone over the barycentric subdivision of the join of the family X.

The complex CX carries the action of the group Π_{v∈V} G_{v}, induced from actions of the groups G_{v} on the sets X_{v} (from the left).

3.5 Proposition. The group G^{c} is isomorphic to the group Π_{v∈V} G_{v}, and the action of G^{c} on the associated complex D_{G^{c}} is equivariantly isomorphic to the action of Π_{v∈V} G_{v} on CX.

Proof: The first assertion is clear. To prove the second assertion we will construct a simplicial isomorphism c : D_{G^{c}} → CX as required, defining it first at vertices. Let [g, S] ∈ D_{G^{c}} be a vertex, where g = Πg_{v} ∈ ΠG_{v}, g_{v} ∈ G_{v}, and S ⊂ V. Put

c_{0}([g, S]) := {g_{v}A_{v} : v ∈ V \ S}

and notice the following properties:

(1) for any vertex [g, S] of D_{G^{c}} its image c_{0}([g, S]) is a well defined vertex in CX;

(2) c_{0} defines a bijection between the vertex sets of the complexes D_{G^{c}} and CX;

(3) both c_{0} and c_{0}^{−1} preserve the adjacency relation on the vertex sets in the corresponding complexes (where two vertices are called adjacent when they span a 1-simplex). Note that, by definition, both complexes D_{G^{c}} and CX have the following property:

each set of pairwise adjacent vertices in the complex spans a simplex of this complex (complexes satisfying this property are often called flag complexes). This property, together with properties (2) and (3) above, implies that the map c_{0} induces a simplicial isomorphism c : D_{G^{c}} → CX.

Now, if g′ = Πg′_{v} ∈ G^{c} = ΠG_{v}, with g′_{v} ∈ G_{v}, we have

g′ · c([g, S]) = g′ · {g_{v}A_{v} : v ∈ V \ S} = {g′_{v}g_{v}A_{v} : v ∈ V \ S} = c([g′g, S]) = c(g′ · [g, S]),

and hence c is equivariant. This finishes the proof.

3.6 Alternative description of Δ_{G}. Denote by Q the quotient of the action of Π_{v∈V} G_{v} on CX, and by q : CX → Q the associated quotient map. Q is easily seen to be the simplicial cone over the barycentric subdivision of the simplex spanned by the indexing set V of the family X. Observe now that the equivariant isomorphism c : D_{G^{c}} → CX of Proposition 3.5 induces an isomorphism ε : P^{c} → Q of the quotients, and thus we clearly have q ∘ c = ε ∘ π^{c}. In fact, ε describes at vertices by ε(S) = V \ S. Define the map δ : P → Q by δ := ε ∘ p^{c}. Proposition 3.5 implies then the following.

3.6.1 Corollary. The subcomplex q^{−1}(δ(P)) ⊂ CX is invariant under the action of the group Π_{v∈V} G_{v} and the action of this group restricted to this subcomplex is equivariantly isomorphic to the action of G^{c} on Δ_{G}.

Slightly departing from the main topic of the paper, we give the following interesting consequence of Theorem 3.3.

3.7 Corollary. A graph product (along any finite graph) of pairs (G_{v}, A_{v}) is virtually torsion free iff all G_{v} are virtually torsion free.

Proof: Since the groups G_{v} inject into the graph product G = Π_{Γ}(G_{v}, A_{v}), they are clearly all torsion free if their graph product is. To prove the converse, observe that by Theorem 3.3, G is a semidirect product of the group G^{c} = Π_{v∈V} G_{v} by the fundamental group π_{1}(Δ_{G}). Since the space Δ_{G} is finite dimensional and aspherical (its universal cover Δ̃_{G} is isomorphic with the Davis’ realization of a building, and hence contractible, see [D]), its fundamental group is torsion free, and the corollary follows.

4. Large common subgroups and the proof of Theorem 1.

4.1. Subgroups. Let (G_{v}, A_{v}) and (G_{v}^{∗}, A_{v}^{∗}) be two families of pairs of groups. Denote by G and G^{∗} the corresponding graph products of pairs along the same graph Γ, and by G^{c} and (G^{∗})^{c} the corresponding graph products along the complete graph Γ^{c}. Let ω^{c} : G → G^{c} and (ω^{∗})^{c} : G^{∗} → (G^{∗})^{c} be the homomorphisms induced by functoriality from the inclusion map Γ → Γ^{c}.

For each v ∈ V let H_{v} < G_{v} and H_{v}^{∗} < G_{v}^{∗} be arbitrary subgroups. Denote by H and H^{∗} preimages of subgroups ΠH_{v} < ΠG_{v} = G^{c} and ΠH_{v}^{∗} < ΠG_{v}^{∗} = (G^{∗})^{c} under the maps ω^{c} and (ω^{∗})^{c} respectively.

4.2. Theorem. If the left actions of H_{v} on G_{v}/A_{v} and of H_{v}^{∗} on G_{v}^{∗}/A_{v}^{∗} are equivariantly isomorphic for all v ∈ V, then the actions of H on D_{G} and of H^{∗} on D_{G^{∗}} are equivariantly isomorphic. In particular, the subgroups H and H^{∗} are isomorphic.

Proof: Let X and X^{∗} be the families of the sets of cosets for the families (G_{v}, A_{v}) and (G_{v}^{∗}, A_{v}^{∗}) respectively. Under assumptions of the theorem, the actions of products ΠH_{v} on CX and ΠH_{v}^{∗} on CX^{∗} are equivariantly isomorphic. Applying Corollary 3.6.1, we conclude that the actions of the groups ΠH_{v} and ΠH_{v}^{∗} on the complexes Δ_{G} and Δ_{G^{∗}} respectively, are equivariantly isomorphic.

Denote by H̃ and H̃^{∗} the preimages of the products ΠH_{v} and ΠH_{v}^{∗} by the homomorphisms G̃ → ΠG_{v} and G̃^{∗} → ΠG_{v}^{∗} respectively. It follows that the actions of H̃ on Δ̃_{G} and of H̃^{∗} on Δ̃_{G^{∗}} are equivariantly isomorphic. But, due to Theorem 3.3, these actions are equivariantly isomorphic to the actions of H on D_{G} and of H^{∗} on D_{G^{∗}} respectively, hence the theorem.

4.3. Corollary. Let (G_{v}, A_{v}) and (G_{v}^{∗}, A_{v}^{∗}) be two families of group pairs indexed by the vertex set V of a finite graph Γ. Suppose that for all v ∈ V there exist subgroups H_{v} < G_{v} and H_{v}^{∗} < G_{v}^{∗} of finite index, such that the left actions of H_{v} on G_{v}/A_{v} and of H_{v}^{∗} on G_{v}^{∗}/A_{v}^{∗} are equivariantly isomorphic. Then the graph products G = Π_{Γ}(G_{v}, A_{v}) and G^{∗} = Π_{Γ}(G_{v}^{∗}, A_{v}^{∗}) are commensurable.

Proof: According to Theorem 4.2 the groups G and G^{∗} share a subgroup H = H^{∗}, which is of finite index in both of them.

4.4. Proof of Theorem 1. Under assumptions of Theorem 1, the left actions of the group H_{v} on G_{v} and on G_{v}^{∗} are clearly equivariantly isomorphic. Then by Corollary 4.3, the graph products Π_{Γ}G_{v} and Π_{Γ}G_{v}^{∗} share a subgroup H, which is easily seen to be of index Π_{v∈V} [G_{v} : H_{v}] in both graph products.

5. Comments, examples and applications.

5.1 Is strong commensurability a necessary assumption in Theorem 1? Considering free products Z_{2} ∗ Z_{2} and Z_{3} ∗ Z_{3} shows that one needs a hypothesis

stronger than commensurability to guarantee commensurability of graph products. A more delicate example is provided by a family of graph products along the pentagon, where at each vertex we put the group Z_{p}. Bourdon computes in [B1] an invariant (conformal dimension at infinity) of the hyperbolic groups arising in this way. His invariant shows that as p varies, the graph products are not even quasiisometric, hence noncommensurable.

A more subtle reason for noncommensurability occurs for free products of surface groups. According to Whyte [W], the groups M_{g} ∗ M_{g} and M_{h} ∗ M_{h} are quasiisometric if g, h ≥ 2. On the other hand, we have the following well known fact.

5.1.1 Lemma. Free products M_{g} ∗ M_{g} and M_{h} ∗ M_{h} of surface groups are not commensurable if g ≠ h.

Proof: Recall that Kurosh theorem asserts that if N is a subgroup of finite index i in M_{1} ∗ M_{2}, then N is a free product

N_{1} ∗ N_{2} ∗ … ∗ N_{k} ∗ F_{l},

where each N_{j} is a subgroup of finite index in either M_{1} or M_{2}, F_{l} is a free group of rank l, and moreover i = k + l − 1. Now assume M_{1}, M_{2} are fundamental groups of orientable aspherical manifolds of the same dimension m (e.g. surface groups). One readily sees that k = b^{m}(N) = rankH^{m}(N, Z) while l is the rank of the kernel in H^{1}(N, Z) of the cup product H^{1}(N, Z) × H^{m−1}(N, Z) → H^{m}(N, Z), interpreted as a bilinear form. Hence if one knows N, one knows the index of N as a subgroup in M_{1} ∗ M_{2}. This implies that if the free products M_{1} ∗ M_{2} and M′_{1} ∗ M′_{2} of two such group pairs are commensurable, they are strongly commensurable.

Now, if g ≠ h then the groups M_{g} ∗ M_{g} and M_{h} ∗ M_{h} are not strongly commensurable, because they have different Euler characteristics. It follows that these groups are not commensurable.

5.2 Commensurability of graph products as transformation groups.

As it is shown in Section 1, to each graph product G of group pairs there is associated a right angled building D_{G} on which G acts canonically by automorphisms. Such buildings corresponding to different groups G may sometimes be isomorphic. In particular, we have

5.2.1 Lemma. Let (G_{v}, A_{v})_{v∈V} and (G_{v}^{∗}, A_{v}^{∗})_{v∈V} be two families of groups and subgroups, indexed by a finite set V. Suppose that for each v ∈ V the indices (not necessarily finite) [G_{v} : A_{v}] and [G_{v}^{∗} : A_{v}^{∗}] are equal. Then for any graph Γ on the vertex set V the buildings D_{G} and D_{G^{∗}} associated to the graph products G = Π_{Γ}(G_{v}, A_{v}) and G^{∗} = Π_{Γ}(G_{v}^{∗}, A_{v}^{∗}) are isomorphic.

Proof: Observe that, under assumptions of the lemma, the complexes D_{G^{c}} and D_{(G^{∗})^{c}}, and hence also their subcomplexes Δ_{G} and Δ_{G^{∗}}, are isomorphic. Since by Theorem 3.3 the buildings D_{G} and D_{G^{∗}} are the universal covers of the complexes Δ_{G} and Δ_{G^{∗}}, the lemma follows.

Call two graph products commensurable as transformation groups if their associated buildings are isomorphic and if they contain subgroups of finite index whose actions on the corresponding buildings are equivariantly isomorphic. The arguments we give in this paper show that the graph products satisfying our assumptions are not only commensurable but also commensurable as transformation groups (see Theorem 4.2). Closer examination of these arguments shows that the strong commensurability condition of Theorem 1 (and a more general condition of Corollary 4.3) is not only sufficient, but also necessary for two graph products of groups (of group pairs respectively) to be commensurable as transformation groups. The details of this argument are not completely immediate, but we omit them.

5.3 Special cases of Theorem 1. Theorem 1 has interesting special cases resulting from various examples of strongly commensurable groups. The simplest class of examples is given by finite groups of equal order. Thus

5.3.1 Corollary. Let (G_{v})_{v∈V} and (G_{v}^{∗})_{v∈V} be two families of finite groups indexed by the vertex set V of a finite graph Γ. Suppose that for each v ∈ V we have |G_{v}| = |G_{v}^{∗}|. Then the graph products Π_{Γ}G_{v} and Π_{Γ}G_{v}^{∗} are strongly commensurable.

The infinite cyclic group Z and the infinite dihedral group D_{∞} are strongly commensurable, since they both contain an infinite cyclic subgroup of index two. Thus a graph product of infinite cyclic groups is commensurable with the corresponding graph product of infinite dihedral groups. Since D_{∞} = Z_{2} ∗ Z_{2}, it follows from 1.3 that any graph product of infinite dihedral groups is a right angled Coxeter group. Graph products of infinite cyclic groups are the so called right angled Artin groups, and thus we reprove the following result from [DJ]:

5.3.2 Corollary. Right angled Artin groups are commensurable with right angled Coxeter groups.

A source of strongly commensurable groups is given by subgroups of the same finite index in some fixed group. The intersection of two such subgroups has clearly the same finite index in both of them. As an example of this kind consider a natural number g ≥ 2 and a tessellation of the hyperbolic plane H^{2} by regular 4g-gons with all angles equal to π/2g (so that 4g tiles meet at each vertex). Let T be the group of all symmetries of this tessellation, and W_{g} < T the Coxeter group generated by reflections in sides of a fixed 4g-gon. Consider also the fundamental group M_{g} of the closed surface of genus g and note that this group can be viewed as subgroup of the group T. Since the groups W_{g} and M_{g} have the same fundamental domain in H^{2} (equal to a single 4g-gon), they have clearly the same index in T (equal to 8g, the number of symmetries of a 4g-gon), and hence are strongly commensurable. By the fact that graph products of Coxeter groups are again Coxeter groups, Theorem 1 implies the following.

5.3.3 Corollary. Graph products of surface groups are commensurable with Coxeter groups.

Pairs of subgroups of the same finite index in a given group (being thus strongly commensurable) are applied also in the following.

5.3.4 Proposition. Graph products of arbitrary subgroups of finite index in right angled Coxeter groups are commensurable with right angled Coxeter groups.

Proof: In view of the fact that graph products of right angled Coxeter groups remain in this class, it is sufficient to show that each finite index subgroup in a right angled Coxeter group W is strongly commensurable with another right angled Coxeter group. This is clearly true for finite groups, as they are (both groups and subgroups) isomorphic to products of the group Z_{2}. To prove this for an infinite group W, we will exhibit in W a family W_{n} : n ∈ N of subgroups, indexed by all natural numbers, with [W : W_{n}] = n, such that each of the groups W_{n} is also a right angled Coxeter group.

Note that if W is infinite, it contains two generators t and s whose product ts has infinite order in W. Let D be a fundamental domain in the Coxeter-Davis complex Σ of W. D is a subcomplex in Σ with the distinguished set of ”faces”, so that reflections with respect to those faces are the canonical generators of W. Since the faces of the reflections t and s are disjoint, the following complex

D_{n} := \{ \begin{matrix}D ∪ tD ∪ stD ∪ … ∪ (st)^{k}D & if n = 2k + 1 \\ D ∪ tD ∪ stD ∪ … ∪ t(st)^{k−1}D & if n = 2k \end{matrix}

is a fundamental domain of a subgroup W_{n} < W generated by reflections with respect to ”faces” of this complex. Since by comparing fundamental domains we get [W : W_{n}] = n, the proposition follows.

The example discussed just before Corollary 5.3.3 generalizes as follows. Let (T_{v})_{v∈V} be a family of topological groups and let Λ_{v} ⊂ T_{v} and Λ_{v}^{∗} ⊂ T_{v} be two families of lattices such that volumes of the quotients T_{v}/Λ_{v} and T_{v}/Λ_{v}^{∗} are finite and equal for all v. Suppose also that for each v ∈ V there is t ∈ T_{v} such that the intersection t^{−1}Λ_{v}t∩Λ_{v}^{∗} has finite index in both Λ_{v}^{∗} and the conjugated lattice t^{−1}Λ_{v}t. Then for each v the lattices Λ_{v} and Λ_{v}^{∗} are strongly commensurable, and hence the graph products Π_{Γ}Λ_{v} and Π_{Γ}Λ_{v}^{∗} are commensurable for any graph Γ with the vertex set V.

For surface groups commensurability condition is a very weak one and we have the following

5.3.5 Fact. Let M and N be two 2-dimensional orbifolds which are developable. Then their fundamental groups G_{M} and G_{N} are strongly commensurable iff the orbifold Euler characteristics of M and N are equal.

Clearly, Fact 5.3.5 allows to formulate the appropriate result on commensurability of graph products of 2-orbifold groups. On the other hand, combining this fact with Theorem 1 and with the argument based on Kurosh’ theorem (as in the proof of Lemma 5.1.1) one gets

5.3.6 Corollary. Under assumptions and notation of Fact 5.3.5, the free products G_{M} ∗ G_{M} and G_{N} ∗ G_{N} are commensurable iff the orbifold Euler characteristics of M and N are equal.

We now pass to applications that require the full strength of Corollary 4.3 rather than that of Theorem 1.

5.4 Orthoparabolic subgroups of Coxeter groups. Recall that parabolic subgroup of a Coxeter group W is any group generated by a subset S′ in the canonical generating set S for W. An orthoparabolic subgroup of a Coxeter group W is a normal subgroup J = ker ρ for a homomorphism ρ : W → P to a parabolic subgroup P such that ρ|_{P} = id_{P}. We say that P is the orthogonal parabolic of J. Note that a homomorphism ρ as above, and hence also an orthoparabolic subgroup orthogonal to P, not always exists.

Since the left actions of a group J on itself and on the cosets W/P are equivariantly isomorphic, Theorem 4.2 implies

5.4.1 Corollary. If for each v ∈ V group J_{v} is an orthoparabolic subgroup in a Coxeter group W_{v}, orthogonal to a parabolic subgroup P_{v}, then the graph product Π_{Γ}J_{v} is a subgroup in the graph product Π_{Γ}(W_{v}, P_{v}). This subgroup has finite index iff the subgroups P_{v} are finite for all v ∈ V.

Applying presentations from 1.4, note that any graph product Π_{Γ}(W_{v}, P_{v}) of pairs of a Coxeter group and its parabolic subgroup is again a Coxeter group. Thus Corollary 5.4.1 implies

5.4.2 Corollary. A graph product of orthoparabolic subgroups in Coxeter groups is a subgroup in a Coxeter group. If the orthogonal parabolic subgroups are all finite, it is a finite index subgroup in a Coxeter group.

Finite cyclic groups Z_{p} are orthoparabolic in the dihedral groups D_{p} (as well as Z in D_{∞}). This again allows to reprove (and extend) the result of [DJ] (compare 5.3.2 above):

5.4.3 Corollary. Graph products of cyclic groups (among them right angled Artin groups) are subgroups of finite index in Coxeter groups.

More generally, an even subgroup of a Coxeter group is the kernel of the homomorphism h : W → Z_{2} which sends all generators of W to the generator of Z_{2}. For example, triangle groups T (p, q, r) and other rotation groups of some euclidean or hyperbolic tessellation are the even subgroups of the Coxeter reflections groups related to these tessalation. Since these groups are clearly orthoparabolic we have

5.4.4 Corollary. Graph products of even subgroups of Coxeter groups are finite index subgroups in Coxeter groups.

Although it is fairly hard to find orthoparabolics in general Coxeter groups, they are plentiful in right angled groups, or more generally in groups where all entries of the Coxeter matrix are even. There, for every parabolic subgroup there exist orthogonal to it orthoparabolics (usually many different ones).

5.5 Graph products of finite group pairs.

Note first that combining Corollaries 5.4.3 and 5.3.1 we obtain

5.5.1 Corollary. Graph products Π_{Γ}G_{v} of finite groups G_{v} are commensurable with Coxeter groups.

Next, applying Corollary 4.3 with trivial groups H_{v}, we get

5.5.2 Corollary. Graph products Π_{Γ}(G_{v}, A_{v}) and Π_{Γ}(G_{v}^{∗}, A_{v}^{∗}) of finite group pairs are commensurable if the equalities [G_{v} : A_{v}] = [G_{v}^{∗} : A_{v}^{∗}] hold for all v ∈ V.

An argument referring to above corollaries and using cyclic groups of orders [G_{v} : A_{v}] proves then the following.

5.5.3 Corollary. Graph products of finite group pairs are commensurable with Coxeter groups.

In the rest of this subsection we prove the following slightly stronger result, under slightly stronger hypotheses:

5.5.4 Proposition. Let (G_{v}, A_{v})_{v∈V} be a family of pairs of a finite group and its subgroup. Suppose that the left action of G_{v} on the cosets G_{v}/A_{v} is effective for each v ∈ V. Then any graph product Π_{Γ}(G_{v}, A_{v}) is a subgroup of finite index in a Coxeter group.

Proof: Canonical action of each of the groups G_{v} on the cosets G_{v}/A_{v} defines a homomorphism i_{v} : G_{v} → S_{G_{v}/A_{v}} = S_{|G_{v}/A_{v}|} to the symmetric group on the set of cosets. By the assertion of the proposition, this homomorphism is injective. Consider a subgroup Stab(A_{v}, S_{G_{v}/A_{v}}) = S_{|G_{v}/A_{v}|−1} and note that i_{v}(A_{v}) ⊂ Stab(A_{v}, S_{G_{v}/A_{v}}). It follows that there is a homomorphism i : Π_{Γ}(G_{v}, A_{v}) → Π_{Γ}(S_{G_{v}/A_{v}}, Stab(A_{v}, S_{G_{v}/A_{v}})) = Π_{Γ}(S_{|G_{v}/A_{v}|}, S_{|G_{v}/A_{v}|−1}) between the graph products. Now, for each v ∈ V the action of G_{v} on G_{v}/A_{v} is easily verified to be equivariantly isomorphic (by i_{v}) to the action of the image group i_{v}(G_{v}) on the cosets S_{G_{v}/A_{v}}/Stab(A_{v}, S_{G_{v}/A_{v}}). It follows from Theorem 4.2 that the homomorphism i is injective and it maps the graph product Π_{Γ}(G_{v}, A_{v}) to the subgroup of finite index in the graph product Π_{Γ}(S_{|G_{v}/A_{v}|}, S_{|G_{v}/A_{v}|−1}).

Symmetric group S_{|G_{v}/A_{v}|} is a Coxeter group and its subgroup S_{|G_{v}/A_{v}|−1} is a parabolic subgroup. By the remark before Corollary 5.4.2, a graph product of symmetric group pairs is a Coxeter group, and thus the proposition follows.

5.5.5 Remark. Removing in Proposition 5.5.4 the assumption of effectiveness for the actions of G_{v} on G_{v}/A_{v} one can get a similar conclusion for the reduced graph products of pairs (G_{v}, A_{v}) as defined in 2.2.

5.6 Groups of automorphisms of locally finite buildings.

It is an open question (except in dimension 1, [L]) whether any two groups of automorphisms acting properly discontinuously and cocompactly on a fixed locally finite right angled buildings are commensurable as transformation groups. The building D_{G} associated to a graph product G = Π_{Γ}(G_{v}, A_{v}) is locally finite iff the indices [G_{v} : A_{v}] are finite for all v ∈ V. The action of G on D_{G} is then properly discontinuous iff the groups G_{v} are all finite. Furthermore, since we always assume that Γ is finite, this action is automatically cocompact.

We may now ask above question in the restricted class of appropriate graph products. By using Lemma 5.2.1 and Corollary 5.5.2 we get

5.6.1 Corollary. Let G = Π_{Γ}(G_{v}, A_{v}) and G^{∗} = Π_{Γ}(G_{v}^{∗}, A_{v}^{∗}) be two graph products of finite group pairs along the same graph Γ. Suppose that for each v ∈ V we have |G_{v}| = |G_{v}^{∗}| and |A_{v}| = |A_{v}^{∗}|. Then the associated buildings D_{G} and D_{G^{∗}} are locally finite and isomorphic, and the actions on them are properly discontinuous and cocompact. Moreover, the groups G and G^{∗} are commensurable as transformation groups.

5.6.2 Remark. By looking more closely one can show that the assumptions of Lemma 5.6.1 are necessary for the buildings D_{G} and D_{G^{∗}} to be locally finite and isomorphic, and to carry properly discontinuous actions of G and G^{∗}. Thus we may conclude that the question 5.6 has positive answer in the class of (associated actions of) graph products. We omit the details of the argument.

5.7 Linearity of graph products. In [DJ] it was pointed out that commensurability of right angled Artin groups (i.e. graph products of infinite cyclic groups) and right angled Coxeter groups implies linearity of the former: Coxeter groups are linear, and groups commensurable with linear groups are linear by inducing representation. By the same argument we get that graph products of groups from various other classes are linear. For example, Corollaries 5.3.3 and 5.5.3 imply the following.

5.7.1 Corollary. Graph products of surface groups and graph products of pairs of finite groups are linear.

5.7.2 Remark. Bourdon [B2] using an entirely different method constructed faithful linear representations of certain graph products of cyclic groups. The target of any of his representations is the Lorenz group SO(N, 1) and the dimension is much smaller than of ones constructed for that group using Corollary 5.7.1.

Without referring to commensurability we get that graph products of any subgroups in Coxeter groups are linear. This follows from the fact that graph products of Coxeter groups are Coxeter groups. The similar fact for pairs of Coxeter groups and their parabolic subgroups implies

5.7.3 Corollary. Let (W_{v}, P_{v}) be a family of pairs where W_{v} are Coxeter groups and P_{v} are their their parabolic subgroups. For each v ∈ V let H_{v} be a subgroup of W_{v}. Then any graph product of the family of pairs (H_{v}, H_{v} ∩ P_{v}) is a linear group.

Proof. A graph product Π_{Γ}(H_{v}, H_{v} ∩ P_{v}) is a subgroup of Π_{Γ}(W_{v}, P_{v}), which is a Coxeter group.

It is very likely that graph products of linear groups are linear.

Bibliography

[BH] M. Bridson, A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer, 1999.

[B1] M. Bourdon, Sur les immeubles fuchsiennes et leur type de quasi-isométrie, Ergod. Th. and Dynam. Sys. 20 (2000), 343-364.

[B2] M. Bourdon, Sur la dimension de Hausdorff de l’ensemble limite d’une familie de sous-groupes convexes co-compactes, C. R. Acad. Sci. Paris, t. 325, Serie I (1997), 1097-1100.

[D] M. Davis, Buildings are CAT (0), in: Geometry and cohomology in group theory (Durham 1994), Cambridge UP, 1998.

[DJ] M. Davis, T. Januszkiewicz, Right angled Artin groups are commensurable with right angled Coxeter groups, J. of Pure and Appl. Algebra 153 (2000), 229-235.

[L] F. T. Leighton, Finite common coverings of graphs, J. Comb. Theory (Ser. B) 33 (1982), 231-238.

[W] K. Whyte, Amenability, Bilipschitz Equivalence, and the Von Neumann Conjecture, arXiv:math.GR/970402.

Tadeusz Januszkiewicz & Jacek Świa̧tkowski,

Instytut Matematyczny UWr, pl Grunwaldzki 2/4; 50-384 Wrocław, Poland

    tjan@math.uni.wroc.pl                         swiatkow@math.uni.wroc.pl