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Asymptotic properties of groups acting on complexes

We examine asymptotic dimension and property A for groups acting on complexes. In particular, we prove that the fundamental group of a finite, developable complex of groups will have finite asymptotic dimension provided the geometric realization of the development has finite asymptotic dimension and the vertex groups are finitely generated and have finite asymptotic dimension. We also prove that property A is preserved by this construction provided the geometric realization of the development has finite asymptotic dimension and the vertex groups all have property A. These results naturally extend the corresponding results on preservation of these large-scale properties for fundamental groups of graphs of groups. We also use an example to show that the requirement that the development have finite asymptotic dimension cannot be relaxed.


Introduction
The asymptotic approach to finitely generated groups became popular following the work of Gromov [9]. In his study of asymptotic invariants of finitely generated groups, Gromov defined asymptotic dimension (asdim), the large-scale analog of Lebesgue covering dimension. G. Yu [16] applied asymptotic dimension to the Novikov higher signature conjecture for groups, showing that the conjecture holds for groups with finite asymptotic dimension. Later, Yu [17] defined another asymptotic invariant for discrete metric spaces and finitely generated groups called property A. This is a weak form of amenability which also implies the Novikov conjecture for groups. (For an introduction to the Novikov and related conjectures, see [8]. ) We wish to consider finitely generated groups as metric spaces. Let Γ be a finitely generated group with generating set S = S −1 . The S-norm on Γ is the norm given by setting γ S = 0 precisely when γ is the group identity and otherwise taking γ S to be the minimal length of any S-word presenting the element γ. Then, one can define the (left-invariant) word metric associated to S by dist S (g, h) = g −1 h S . The metrics corresponding to two finite generating sets S and S ′ are Lipschitz equivalent. Asymptotic dimension and property A are invariants of Lipschitz equivalent metric spaces, so these properties are intrinsic to the group Γ and not the metric space associated to a specific generating set.
In view of Yu's results, [16], [17], it is important to know which groups have finite asdim or property A. Gromov [9] showed that hyperbolic groups have finite asdim . Dranishnikov and Januszkiewicz proved in [6] that Coxeter groups have finite asdim . Dranishnikov and the author proved that the finiteness of asdim is preserved by the amalgamated free product and HNN extension, and more generally any fundamental group of a finite graph of groups with vertex groups having finite asdim, see [2], [3]. Higson and Roe [12] showed that finitely generated groups with finite asymptotic dimension have property A. In [15], J.-L. Tu proved that property A is preserved by the fundamental group of a finite graph of groups where the vertex groups all have property A. At first it was not know whether there could be a finitely generated group with infinite asdim or which does not have property A. A recent example of such a group due to Gromov [10] and Dranishnikov, Gong, Lafforgue and Yu [7] , have made determining precisely which groups have these properties interesting.
The Bass-Serre theory of graphs of groups generalizes the constructions of amalgamated free products and HNN extensions (see [14]). There is a direct correspondence between groups acting without inversion on trees and fundamental groups of graphs of groups. Complexes of groups were introduced by Haefliger [11] in order to describe actions of groups on simply connected simplicial complexes in the same way that graphs of groups describe the action of groups on trees. The problem that arises is that the quotient of a simplicial complex by a simplicial action may identify faces of simplices. So it may not be the case that the quotient is a simplicial complex. This problem is avoided by introducing combinatorial substitutes for simplicial complexes called small categories without loops (scwols).
In the second section we develop the necessary theory of complexes of groups following [4]. We define scwols, group actions on scwols, complexes of groups, developability of complexes of groups, and the associated fundamental group of a complex of groups.
In the third section, we define the R-stabilizer which plays the rôle of the stabilizer in the study of the large-scale properties of groups acting on metric spaces. We also characterize the R-stabilizers for groups acting on scwols.
In the fourth section, we define asdim and obtain our main result on asymptotic dimension: Theorem. Let G(Y) be a developable complex of groups over a finite scwol Y with development X such that asdim X < ∞, and such that every local group G σ has asdim G σ < ∞. If π 1 denotes the fundamental group of the complex of groups, we have asdim π 1 < ∞.
In the fifth section, we define property A and prove an analogous result for property A: Theorem. Let G(Y) be a developable complex of groups over a finite scwol Y with development X such that asdim X < ∞, and such that every local group G σ has property A. If π 1 denotes the fundamental group of the complex of groups, then π 1 has property A.
In the final section, we use an example of a finitely generated group which does not have either property A or finite asdim to show that the results we obtain here cannot be improved by relaxing the condition that the development have finite asdim .

Complexes of Groups
Our notation and development will follow Bridson-Haefliger [4].
Definition. A small category without loops (abbreviated scwol) is a set X which is the disjoint union of a vertex set V (X ) and an edge set E(X ). There are maps which assign to each edge a the initial vertex of a and the terminal vertex of a, respectively. Let There is also a map Let E (k) (X ) denote the composable sequences of edges of length k, i.e., We define the dimension of the scwol X to be the maximum k such that E (k) (X ) is not empty.
Definition. The geometric realization |X | is a piecewise Euclidean polyhedral complex, with each k-cell isometric to the standard simplex ∆ k . There is one such ksimplex A for each A ∈ E (k) (X ). The identifications are the obvious ones, induced by the face relation among simplices.
Observe that the geometric realization need not be a simplicial complex, since it may be the case that the intersection of two simplices is a union of faces. One can eliminate this problem by taking the barycentric subdivision, if one requires simplicial complexes. The geometric dimension of |X | is the same as the dimension of the combinatorial object X .

Definition. A group action on a scwol is a homomorphism G → Aut(X ) satisfying
(1) for every g ∈ G, and for all a ∈ E(X ) g.i(a) = t(a).
(2) for every g ∈ G, and for all a ∈ E(X ) if g.i(a) = i(a), then g.a = a.
Notice that a group action on a scwol induces an isometric action of the group on the geometric realization |X |. Since we are primarily concerned with isometric actions on metric spaces, this is the action that we consider.
One forms the quotient Y = G\X of the scwol X by the action of G by taking V (Y) = G\V (X ), and E(Y) = G\E(X ). One can verify that Y has the structure of a scwol.
, there corresponds a group G σ called the local group at σ; (2) for each a ∈ E(Y) there exists an injective homomorphism ψ a : G i(a) → G t(a) ; and Given a group G and an action of G on the scwol X , there is an explicit construction of the complex of groups over the quotient scwol which we do not describe here. However, on the other hand, it is not always the case that an arbitrary complex of groups can be associated to a group action on some scwol X . When this occurs, we say that the complex of groups is developable and we refer to the associated scwol X as the development of G(Y).
It is clear that scwols of dimension 1 must have precisely two types of vertices: sources and sinks. A source is an initial vertex of every edge it is contained in and a sink is a terminal vertex of every edge it is contained in. Every one-dimensional simplicial complex (graph) can be given the structure of a one-dimensional scwol by placing a source vertex in the middle of every edge, thus giving the original vertices the structure of sinks. It is easy to verify that the theory of complexes of groups over one-dimensional scwols is precisely the same as the theory of graphs of groups. Phrased in terms of the language of complexes of groups, the Bass-Serre structure theorem for groups acting without inversion on graphs says that if dim(Y) = 1, then G(Y) is always developable.
When a complex of groups is developable, there is an explicit method of constructing both the scwol X and the group G which acts on the scwol. The scwol X on which the group acts is simply connected and has an explicit description in a similar way to the construction of the treeX in the theory of graphs of groups (see [14]).
Indeed, if G(Y) is a developable complex of groups, then we can define the development D(Y) to be the scwol whose vertices and edges are given by Then the group G acts on the development D(Y) by left multiplication. The development is isomorphic to the scwol X , mentioned above. (See [4] for more details. ) We describe the fundamental group of the complex of groups π 1 (G(Y)) which is the group G, up to isomorphism. As in the theory of graphs of groups, there are two equivalent descriptions of the fundamental group. Both rely on the construction of the auxiliary group F G(Y). Let E ± (Y) denote the collection of symbols {a + , a − } where a ∈ E(Y). The elements of E ± (Y) can be thought of as oriented edges. If e = a + , then define i(e) = t(a) and t(e) = i(a). Accordingly, if e = a − , define t(e) = t(a) and i(e) = i(a). Then, define F G(Y) to be the free product of the local groups G σ and the free group generated by the collection E ± (Y) subject to the additional relations: (1) (a + ) −1 = a − , and (a − ) −1 = a + ; (2) a + b + = g a,b (ab) + ; (3) ψ a (g) = a + ga − , for all g ∈ G i(a) . The first description of the fundamental group is in terms of G(Y)-loops based at some fixed vertex σ 0 . An edge path in Y is a sequence (e 1 , . . . , e k ) with t(e i ) = i(e i+1 ), for all i = 1, . . . , k − 1. By a G(Y)-path issuing from σ 0 we mean a sequence (g 0 , e 1 , g 1 , . . . , e k , g k ), where (e 1 , . . . , e k ) is an edge path in Y, and g 0 ∈ G σ0 , and g i ∈ G t(ei) , for i > 0. We associate the word g 0 e 1 . . . e k g k ∈ F G(Y) to the path described above. A G(Y)-loop based at σ 0 is a G(Y) path with t(e k ) = σ 0 . There is an obvious notion of homotopy on G(Y)-paths, i.e., the notion of homotopy on the geometric realization. The fundamental group π 1 (G(Y), σ 0 ) is the collection of all words associated to G(Y)-loops based at σ 0 , up to homotopy equivalence.
The second description is much simpler. Let T be a maximal tree in |Y| (1) . Then, π 1 (G(Y), T ) is F G(Y) subject to the additional relation a + = 1, for all a ∈ T. For a connected scwol, there is an isomorphism π 1 (G(Y), σ 0 ) → π 1 (G(Y), T ).

Groups Acting on Metric Spaces
For group actions considered on a local scale, the stabilizer plays a key rôle. The corresponding notion for group actions considered in the global sense is that of the R-stabilizer which we define presently.
Definition. Let Γ be a group acting on the pointed metric space (X, x 0 ) by isometries. For every R > 0 define the R-stabilizer of the point In [2], Dranishnikov and the author characterized the R-stabilizers of the action of a fundamental group of a graph of groups on the tree corresponding to its development. The following proposition is a natural generalization of that result. Proposition 1. Let G(Y) be a developable complex of groups. Fix a vertex σ 0 ∈ Y, and consider the action of π 1 (G(Y), σ 0 ) on the simply connected scwol X induced by the complex of groups. Then the R-stabilizer W R (σ 0 ) is the set of all elements of π 1 (G(Y), σ 0 ) with associated path c of length not exceeding R.
Proof. Let γ be a word in π which is reduced and for which the path length is R. Then consider d X (γG σ0 , G σ0 ). The path corresponding to γ lifts to a path in X , so this distance is at most R.
Thus, it remains to show that no element of γG σ0 has length less than R if γ has length R in its most reduced form.
To this end, let xg = g 0 e 1 · · · e R g R g, be a reduced word in xG σ0 . Two of the relations on F G(Y) can affect the path length of a word. In order for the relation (e + i e + i+1 ) = g ei,ei+1 (e i e i+1 ) + to occur in this word, we would need two composable edges, in the sense that t(e i+1 ) = i(e i ), but since by definition, we have t(e i ) = i(e i+1 ) for all edges, we conclude that e −1 i = e i+1 . Thus, the only relation that can occur is the type ψ a (g) = a + ga − .
Suppose that a sequence of this type of relation occurs which transform the word xg into g 0 e 1 · · · g k−1 e k g k g ′ e k+1 g ′ k · · · g ′ R . Here two cases can occur. In the first case, edge e k has positive orientation. Then, e k g k g ′ e k+1 = ψ e k (g k g ′ ). But, in this case, we can obtain a reduction of the original path x since e k ge k+1 = ψ e k (g k ). As the original path was reduced, this cannot occur. The other case is when e k+1 has positive orientation. Then, g k g ′ = ψ e k+1 (h) for some h. But, it must be the case that g ′ was generated by this type of relation, as this is the only type that can occur and involves edges. Thus, g ′ is itself ψ e k+1 (h ′ ), for some h ′ . Thus, g = ψ e k+1 (h(h ′ ) −1 ). This enables a reduction of the original word. Thus, the length of any representative of xG σ0 has length no less than R as desired.

Asymptotic Dimension
Asymptotic dimension was introduced by Gromov [9]. It is the coarse analog of Ostrand's characterization of covering dimension for metric spaces, [13].
Definition. Let X be a metric space. We define the asymptotic dimension (asdim) of X by the following inequality: asdim X ≤ n if for every R > 0 there exist n + 1 families of sets U 0 , . . . , U n which are uniformly bounded, which cover X and which are R-disjoint in the sense that any two distinct sets from the same family are at a distance greater than R from each other. We define asdim X = n if it is the case that asdim X ≤ n, but it is not the case that asdim X ≤ n − 1.
The goal of this section is to see that the finiteness of asdim is preserved by the construction of the fundamental group of a developable complex of groups. This is a natural generalization of the main theorem in [3].
Definition. Let X α be a family of subsets of the metric space X. We say that the family satisfies the inequality asdim X α ≤ n uniformly if for every D > 0 there is a number R > 0 and a collection of D-bounded, R-disjoint families {U α i } so that for each α, {U α i } covers X α . A common example of a family satisfying asdim X α ≤ n uniformly is a family of isometric metric spaces.
The following union theorem appears as Theorem 1 in [2].
Theorem (Union Theorem). Let X = ∪ α F α and asdim F α uniformly. Suppose that for any r there exists a set Y r ⊂ X such that asdim Y r ≤ n and the family As a corollary, we have the following finite union theorem.
Theorem (Finite Union Theorem). Let X = ∪ k i=1 X i be a metric space. Then, asdim X ≤ max{asdim X i | i = 1, . . . k} Let G(Y) be a developable complex of groups, with Y finite and asdim |X | ≤ k. Suppose further that the local groups are finitely generated. Then, the finiteness of Y implies that the fundamental group π is finitely generated. Indeed, if S σ denotes a finite generating set for each local group G σ , then we can consider F G(Y) in the metric obtained from the disjoint union of all the S σ and the set E ± (Y). Thus, the notion of asdim is well-defined for the fundamental group of a complex of groups.
Lemma 2. Let π be the fundamental group of a complex of groups G(Y) where Y is finite, the local groups are finitely generated, and the local groups satisfy asdim G σ ≤ n, then for every R > 0, asdim W R (G σ0 ) ≤ n.
Proof. In Proposition 1 we characterized W R (G σ0 ) as the set of all elements in π with length at most R. Let X denote the development of F G(Y).
In order to apply an inductive argument, we consider a larger set, K ⊂ F G(Y) which is the set of all words in F G(Y) issuing from σ 0 . The group π is a subset of K and the set K acts on X by left multiplication. We show that the R-stabilizer of this action has asdim at most n. It follows then that the R-stabilizer of the action of π on X will also have asdim at most n.
In light of the finite union theorem, in order to show asdim W R (G σ0 ) ≤ n, it suffices to show that the subset K j ⊂ K of reduced words in K with length equal to j has asdim K j ≤ n. Indeed, W R (G σ0 ) ⊂ ∪ R j=0 K j , which is a finite union. We proceed by induction. The case j = 0 is true by assumption since K 0 = G σ0 . Consider the case K j+1 with j ≥ 0. Observe that K j+1 ⊂ ∪ a∈E ± (Y) K j aG t(a) .
The orientation of the edge a is an issue, as it determines whether the group G t(a) is a domain or codomain of the function ψ a . Thus, it is necessary to consider two cases separately.
Suppose first that a has negative orientation. So, we are considering K j a − G t(a) . For every r > 0 let Y r = K j a − N r (ψ a (G i(a) )), where the r-neighborhood is taken in the group F G(Y). Then Y r is coarsely equivalent to K j a − ψ a (G i(a) ). Now we have K j a − ψ a (G i(a) ) = K j G i(a) a − , which is just K j a − . Finally, as K j a − is coarsely equivalent to K j , we have asdim Y r = asdim K j , which by the inductive hypothesis does not exceed n.
Next, decompose the set K j a − G t(a) into families {xa − G t(a) }, where the index runs over all x ∈ K j which do not end with an element g ∈ G i(a) . One can still obtain these elements through the relations of F G(Y). For instance, to obtain xga − g ′ , with x of the required form, g ∈ G i(a) and g ′ in G t(a) , simply take the word xa − ψ a (g)g ′ , which is of the required form. Next, observe that the map G t(a) → xa − G t(a) is an isometry in the (left-invariant) word metric, so the family {xa − G t(a) } has asdim ≤ n, uniformly.
In order to apply the union theorem to this family, it remains to show only that the family {xa − G t(a) \Y r } is r-disjoint. To this end, let xa − z and x ′ a − z ′ be given in different families. Then we compute d( Since z and z ′ lie outside of N r (ψ a (G i(a) )), take z = sψ a (g) and z ′ = s ′ ψ a (g ′ ), where s > r, s ′ > r, and ψ a (g) and ψ a (g ′ ) are in ψ a (G t(a) ). Then, Now, in order for this length to be less than r, a reduction must occur in the middle, so that a + and a − annihilate each other. In order for this to occur, we must have g −1 x −1 x ′ g ′ ∈ G i(a) . Thus, x −1 x ′ ∈ G i(a) . But, this means that xa − G t(a) and x ′ a − G t(a) define the same family. Thus, in the case that the edge has negative orientation, we have asdim K j a − G t(a) ≤ n. Next, we consider the case where the edge a has positive orientation. In this case, K j a + G i(a) = K j ψ a (G i(a) )a + , which is coarsely equivalent to K j . We conclude that asdim K j a + G i(a) = asdim K j ≤ n.
The following result appears as Theorem 2 from [2].
Theorem. Assume that a finitely generated group Γ acts by isometries on a metric space X with a base point x 0 and with asdim X ≤ k. Suppose that asdim W R (x 0 ) ≤ n for all R. Then asdim Γ ≤ (n + 1)(k + 1) − 1.
This estimate on the dimension is far from sharp. It is useful only as a means to prove that asdim Γ < ∞. The exact estimate should be n + k. (See [3] for the proof of the exact formula in the case of groups acting on trees by isometries.) As a consequence of the preceding theorem, we have our main result on asdim .
Theorem 3. Let Γ be the fundamental group of a finite developable complex of groups G(Y) corresponding to an action by isometries on the geometric realization of the scwol X . Suppose that the local groups are finitely generated and that asdim G σ ≤ n. Assume additionally that asdim |X | ≤ k. Then asdim Γ ≤ (n + 1)(k + 1) − 1.
The important result is summarized as a corollary: Corollary. Let Γ be the fundamental group of a finite developable complex of groups G(Y) such that the development X has asdim |X | < ∞, and such that every base group G σ has asdim G σ < ∞. Then, asdim Γ < ∞.

Property A
Property A was introduced by G. Yu, [17]. It is a weak form of amenability which, for groups, implies the existence of a uniform embedding into Hilbert space. Thus, the coarse Baum-Connes conjecture and the Novikov conjecture hold for this group.
Definition. Let X be a metric space. Let P (X) denote the set of probability measures on X in the l 1 metric. The metric space X has property A if there exists a sequence of maps a n : X → P (X) satisfying the following two conditions: (1) for every n there is an R so that for every x, supp a n x ⊂ B R (x), and (2) for every K > 0, lim n→∞ sup d(x,y)<K a n x − a n y 1 = 0.
As mentioned in the introduction finitely generated groups with finite asdim have property A, [12]. Tu proved, [15], that the fundamental group of a finite graph of groups in which each vertex group has property A will have property A. In [1], the author generalized Tu's results to groups acting by isometries on metric spaces with finite asdim .
In particular, the theorem proved in [1] is the following: Theorem. Assume that the finitely generated group Γ acts on the metric space X by isometries. Assume that asdim X ≤ n, and that for every R, the R-stabilizer of a basepoint x 0 ∈ X has property A. Then Γ has property A.
In [1] the author proves a union theorem which is analogous to the union theorem for asdim from §4.
Theorem (Union Theorem). Let X = ∪ α X α where the X α are pairwise isometric and have property A. Suppose further that for every r > 0 there is a set Y r so that {X α \ Y r } is r-disjoint. Then, X has property A.
As a consequence we obtain the finite union theorem.
Theorem (Finite Union Theorem). Let X = ∪ n i=1 X i where the X i all have property A. Then, X has property A.
The analog of Lemma 2 for Property A is the following. The proof follows from the proof of Lemma 2.
Lemma 4. Let G(Y) be a developable complex of groups. Suppose that Y a finite, connected scwol, that the local groups are finitely generated, and that the local groups have property A. Then, the R-stabilizer W R (σ 0 ) for some base vertex σ 0 also has property A.
Applying the theorem from [1] cited above, on groups acting on metric spaces with finite asdim, we obtain the following generalization of Tu's theorem.
Theorem 5. Let G(Y) be a developable complex of groups with corresponding development X and fundamental group π. Suppose that asdim |X | is finite and that the stabilizers of the action have property A. Then, π has property A.

Example
The following example illustrates that one must consider the large-scale structure of the development.
Consider a finitely presented group Γ which does not have finite asdim or property A, see [10], [7]. Since the group is finitely presented, there is a finite complex K so that π 1 (K) = Γ. Thus, by taking the complex of groups with each vertex trivial and the scwol whose geometric realization is equal to the complex K, one obtains the fundamental group of the complex of groups equal to the group Γ. The vertex groups have finite asdim and the group acts on the universal cover, so the complex of groups is developable. The complex K is finite, yet the group Γ does not have finite asdim and does not have property A.