BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS

IGOR MINEYEV

Abstract. For a finitely presentable group G, G is hyperbolic if and only if the map H_{b}^{2}(G, V) → H^{2}(G, V) is surjective for any bounded G-module. We prove the remaining “if” direction. The words “bounded G-module” here can be understood in several different ways. We also consider several ways to define a linear homological isoperimetric inequality.

1. Introduction.

The question of cohomological description of hyperbolicity was considered by S. M. Gersten who proved the following

Theorem 1 (Gersten [3]). The finitely presented groups G is hyperbolic iff H_{(∞)}^{2}(G, ℓ_{∞}) = 0.

Here H_{(∞)}^{n}(G, V) is the ℓ_{∞}-cohomology defined by bounded (not necessarily equivariant) cellular cochains in the universal cover of a K(G, 1) complex with finitely many cells in the dimensions up to n. This theorem was generalized by the author [9] to higher dimensions: if G is hyperbolic then H_{(∞)}^{n}(G, V) = 0 for any n ≥ 2 and any normed vector space V (over ℚ or ℝ).

The bounded cohomology of a group is defined by bounded equivariant cochains in the homogeneous bar construction (see the precise definition in the next section). G. A. Noskov [11, p.1068], using results of B. E. Johnson [8], showed a characterization of amenable groups, which we restate in the following form.

Theorem 2 (Noskov [11]). For a group G the following statements are equivalent.

(a) G is amenable.

(b) H_{b}^{1}(G, V^{∗}) = 0 for any bounded G-module V.

(c) H_{b}^{i}(G, V^{∗}) = 0 for any i ≥ 1 and any bounded G-module V.

The expression “bounded G-module V ” here is what we call a “bounded Banach ℝGmodule”, and V^{∗} is the space conjugate to V.

The result presented in this paper is in a sense an analog of the above two theorems: we characterize hyperbolic groups by bounded cohomology. It was shown in [10] that if G is a hyperbolic group, then the map H_{b}^{i}(G, V) → H^{i}(G, V), induced by inclusion, is surjective for any bounded ℚG-module V and any i ≥ 2 (see the definitions in the next section). The goal of the present paper is to show the converse of this result. We only need to require the group G to be finitely presentable for this converse to be true. If this is the case, then surjectivity of the above maps only in dimension 2 is already sufficient for hyperbolicity. Namely, we prove

Date: July 1999.

Theorem 3. For a finitely presentable group G, the following statements are equivalent.

(a) G is hyperbolic.

(b) The map H_{b}^{2}(G, V) → H^{2}(G, V) is surjective for any bounded G-module V.

(c) The map H_{b}^{i}(G, V) → H^{i}(G, V) is surjective for any i ≥ 2 and any bounded Gmodule V.

Here by a “bounded G-module” we mean one of the six concepts M_{1}(G) − M_{6}(G) described in the next section. (See Theorem 9 for a more precise statement.)

This looks quite interesting that the same property can be characterized by the ℓ_{∞}- cohomology and the bounded cohomology, the two theories which do not seem to have much in common. Also, the characterizations of hyperbolic and amenable groups seem strikingly similar. This similarity may be worth being investigated further.

There are two crucial points in our proof. Firstly, for finitely presentable groups, hyperbolicity is equivalent to the existence of a linear isoperimetric inequality for real 1-cycles. This equivalence was proved by S. M. Gersten. (It follows, for example, from Proposition 3.6, Theorem 5.1, and Theorem 5.7 (stated above) in [3].) This isoperimetric inequality for real cycles is a homological version of the usual combinatorial isoperimetric inequality for loops. We give a direct proof of the fact that the linear isoperimetric inequality for filling (usual) real 1-cycles with summable 2-chains implies hyperbolicity.

Secondly, one needs to pick appropriate coefficients V. We take V to be the space of all boundaries of summable 2-chains in a cell complex with a nice G-action. Similar coefficients (but ℤ-modules rather than ℝ-modules), and the associated with them universal cocycle were used by S. M. Gersten [7, Chapter 12] to introduce ℤ-metabolic (or simply metabolic) groups and to give several equivalent definitions. In particular, metabolic groups are hyperbolic, and it was shown in [9] that the corresponding properties ℚ-metabolic and ℝ-metabolic are actually equivalent to hyperbolicity.

The author is thankful to Steve Gersten and Vladimir Troitsky for helpful discussions, and very much appreciates the hospitality of the University of Melbourne and Max-Planck-Institut für Mathematik in Bonn in 1998-99.

2. Preliminaries.

First we need some definitions. Let 𝔽 stand for either ℚ or ℝ, and 𝔸 stand for ℤ, ℚ, or ℝ.

2.1. Abelian group norms. A \underline{normed abelian group} A is an abelian group with an \underline{abelian group norm} | · | : A → ℝ_{+} satisfying

• |a| = 0 iff a = 0 and

• |a + a′′| ≤ |a| + |a′|. for all a, a′ ∈ A.

2.2. Norms. A \underline{normed vector space} W over 𝔽 is a vector space with a \underline{norm} |·| : W → ℝ_{+} satisfying

• |w| = 0 iff w = 0,

• |w + w′′| ≤ |w| + |w′|, and

• |αw| = |α| · |w| for all w, w′ ∈ W and α ∈ 𝔽.

Of course, each norm on W is an abelian group norm, but not conversely.

2.3. ℓ_{1}-norm. Let a free 𝔸-module M have a preferred basis \{m_{i}, i ∈ I\}. The \underline{ℓ_{1}-norm} | · |_{1} on M (with respect to this basis) is given by

|∑_{i∈I} α_{i}m_{i}|_{1} := ∑_{i∈I} |α_{i}|.

The ℓ_{1}-norm is an abelian group norm. Moreover, if M happens to be a vector space over ℚ or ℝ, then it is a norm.

2.4. ℓ_{∞}-norm. Suppose that (W, | · |_{1}) and (W′, | · |) are normed vector spaces, where W is equipped with the ℓ_{1}-norm | · |_{1} with respect to some preferred basis {w_{i}, i ∈ I}. For a linear map φ : W → W′, the \underline{ℓ_{∞}-norm} of φ, |φ|_{∞}, is the operator norm of φ, i.e. |φ|_{∞} is the smallest number K (possibly infinity) such that |φ(w)| ≤ K|w|_{1} for each w ∈ W. One checks that

|φ|_{∞} = sup_{i∈I} |φ(w_{i})|.

2.5. Cell complexes. By a \underline{cell complex} we mean a combinatorial cell complex, i.e. the one in which the boundary of each cell σ is cellulated and the gluing map of σ restricts to homeomorphisms on the open cells of this cellulation. In particular, each 2-cell can be viewed as a polygon.

We always put the path metric on the 1-skeleta of cell complexes, which is induced by assigning length 1 to each edge. A \underline{cellular ball} of radius r centered at a vertex x is the union of all closed cells whose vertices lie r-close to x.

If X is a cell complex and C_{i}(X, 𝔸) is the space of cellular i-chains, we always give C_{i}(X, 𝔸) the ℓ_{1}-norm with respect to the standard basis consisting of i-cells.

2.6. Filling norm for compactly supported chains. If G is a finitely presentable group, there exists a contractible cell complex X with a free cellular G-action such that the induced action on the 2-skeleton of X is cocompact. In particular, the boundary homomorphism ∂_{2} : C_{2}(X, 𝔸) → C_{1}(X, 𝔸) is bounded (with respect to the ℓ_{1}-norms in the domain and the target).

The \underline{filling norm | · |_{fc,𝔸}} on the space B_{1}(X, 𝔸) of 1-boundaries is given by

|b|_{fc,𝔸} := inf {|a|_{1} | a ∈ C_{2}(X, 𝔸), ∂a = b}.

In other words, | · |_{fc,𝔸} is the abelian group norm induced by the map ∂_{2} : C_{2}(X, 𝔸) → C_{1}(X, 𝔸) on its image. One checks that | · |_{fc,ℤ} is an abelian group norm, and | · |_{fc,ℚ} and | · |_{fc,ℝ} are norms. (The condition “|w| = 0 iff w = 0” holds because | · |_{fc,𝔸} dominates the ℓ_{1}-norm on B_{1}(X, 𝔸).)

2.7. ℓ_{1}-completion. Now we restrict to the real coefficients. For X as above, let C_{i}^{(1)}(X, ℝ) be the completion of C_{i}(X, ℝ) with respect to the ℓ_{1}-norm, i = 1, 2, i.e. it is the set of all (absolutely) summable i-chains in X. The boundary homomorphism ∂_{i} : C_{i}(X, ℝ) → C_{i−1}(X, ℝ), i = 1, 2, is continuous, so it extends to a continuous linear map ˆ∂_{i} : C_{i}^{(1)}(X, ℝ) → C_{i−1}^{(1)}(X, ℝ) which commutes with the G-action.

Denote B_{i}^{(1)}(X, ℝ) := Im ˆ∂_{i+1}, Z_{i}^{(1)}(X, ℝ) := Ker ˆ∂_{i}.

2.8. Filling norm for summable chains. Now we consider “the complete version” of the filling norm. The norm \underline{| · |_{f1,ℝ}} on the space B_{1}^{(1)}(X, ℝ) is given by

|b|_{f1,ℝ} := inf {|a|_{1} | a ∈ C_{2}^{(1)}(X, ℝ), ∂a = b}.

Again, since ˆ∂_{2} : C_{2}^{(1)}(X, ℝ) → C_{1}^{(1)}(X, ℝ) is bounded, |·|_{f1,ℝ} dominates |·|_{1} on B_{1}^{(1)}(X, ℝ) and is therefore a norm indeed.

2.9. Bounded cohomology. An 𝔽G-module is called \underline{bounded} if it is normed as a vector space over 𝔽 and G acts on it by linear operators of uniformly bounded norm. For a bounded 𝔽G-module V, the \underline{bounded cohomology} of G with coefficients in V, \underline{H_{b}^{∗}(G, V)}, is the homology of the cochain complex

0 ⟶ C_{b}^{0}(G, V) ⟶^{δ_{0}} C_{b}^{1}(G, V) ⟶^{δ_{1}} C_{b}^{2}(G, V) ⟶^{δ_{2}} ... ,

where

(1) C_{b}^{i}(G, V) := {α : G^{i+1} → V | α is a bounded G-map}

and the coboundary map δ_{i} is defined by

δ_{i}α([x_{0}, ..., x_{i+1}]) := ∑_{k=0}^{i+1}(−1)^{k}α([x_{0}, ..., x̂_{k}, ..., x_{i+1}]). (2)

Here G^{i+1} is considered with the diagonal G-action by left multiplication, and “bounded” means “has bounded image with respect to the norm on V ”.

Equivalently, C_{b}^{i}(G, V) ⊆ Hom_{𝔽G}(C_{i}(G, 𝔽), V) is the subspace of all 𝔽G-morphisms C_{i}(G, 𝔽) → V, which are bounded as linear maps, where C_{i}(G, 𝔽) is the space of all chains (= finite support functions) G^{i+1} → 𝔽 given the ℓ_{1}-norm with respect to the standard basis G^{i+1}. The coboundary homomorphism δ in C_{b}^{i}(G, V) is the dual of the boundary morphism ∂ in C_{∗}(G, 𝔽). The spaces C_{b}^{i}(G, V) are naturally given the ℓ_{∞}-norm dual to the ℓ_{1}-norm on C_{i}(G, 𝔽).

The definition of bounded cohomology is usually given for 𝔽 = ℝ. We give it both for ℚ and ℝ. This might cause the following problem: if V is a normed real vector space, the bounded cohomology H_{b}^{∗}(G, V) is defined in two different ways, because V may be viewed as a normed rational vector space as well. But it turns out that these two definitions coincide in this case, because formulas (1) and (2) make no use of the field 𝔽.

3. Summable 1-chains.

In this section we prove that certain summable 1-chains on a graph can be approximated in a good way by good compactly supported chains. These results (though not in full generality) will be needed in section 4 for isoperimetric inequalities. Essentially, we generalize the following theorem proved by D. J. Allcock and S. M. Gersten.

Theorem 4 (Theorem 3.3 in [1]). If Γ is a graph and f is a summable real-valued 1-cycle on Γ, then there is a countable coherent family C of simple circuits in Γ and a function g : C → [0, ∞) such that f = ∑_{C} g(c)c.

“Coherent” here essentially means that |f|_{1} = ∑_{C} g(c)|c|_{1}.

Let Γ be a graph. As a part of the structure, we assign an orientaton to each edge of Γ. Therefore “a 1-chain on Γ” is the same thing as “a function on the edges of Γ”. The orientation on Γ determines \underline{the initial vertex} ιe and \underline{the terminal vertex} τe for each edge e. A \underline{directed path} p in Γ is a sequence of edges (e_{1}, ..., e_{n}) with τe_{i} = ιe_{i+1}. The initial vertex ιp of the path p is ιe_{1} and the terminal vertex τp of p is τe_{n}. A directed path (e_{1}, ..., e_{n}) is \underline{simple} if the vertices ιe_{1}, ..., ιe_{n} are all distinct. If T is a set of vertices in Γ, we say that p is a \underline{T -path} if it is a directed path in Γ such that ιp, τp ∈ T or ιp = τp (in other words, the homological boundary of p is supported in T).

Let f be a(n absolutely) summable 1-chain in Γ. For a vertex v in Γ, define

Div_{v}(f) := ∑_{ιe=v} f(e) − ∑_{τe=v} f(e),

where e stands for edges in Γ (with the fixed orientation).

By Γ(f) we denote the same graph Γ but with an orientation on the edges chosen so that f(e) ≥ 0 for each edge e. Let Γ_{+}(f) be the minimal subgraph of Γ(f) containing all the edges e with f(e) ≠ 0 (i.e. f(e) > 0).

Lemma 5. Let Γ be a graph and T be a set of vertices in Γ. If h is a summable real 1-chain such that supp(∂h) ⊆ T and Γ_{+}(h) contains no nontrivial simple T -paths, then h = 0.

Proof. (cf Lemma 3.2 in [1].) Suppose to the contrary that h ≠ 0, i.e. there is an edge e′ in Γ_{+}(h) (hence h(e′) > 0). Let Γ′ be the (non-empty) union of all directed paths (e_{1}, ..., e_{n}) in Γ_{+}(h) with e_{1} = e′. Then e′ is the only edge in Γ′ incident on ιe′ (otherwise there would be a non-trivial directed edge cycle in Γ′ ⊆ Γ_{+}(h).)

\underline{Case 1.} We assume for the moment that Γ′ does not contain vertices from T, except possibly for ιe′. Let h′ be the summable chain in Γ which coincides with h on Γ′ and takes value 0 outside. If v is any vertex in Γ′ different from ιe′, then v ∉ T ⊇ supp(∂f) and also, by the definition of Γ′, Γ′ contains all the edges e of Γ_{+}(h) with ιe = v, so

0 = Div_{v}(h) ≤ Div_{v}(h′).

Obviously, for the vertex ιe′ the strong inequality holds:

0 < h(e′) = h′(e′) = Div_{ιe′}(h′). (3)

Since h′ is (absolutely) summable, rearranging the terms we get

0 ≤ ∑_{v vertex in Γ} Div_{v}(h′) = ∑_{v vertex in Γ} (∑_{ιe=v} h′(e) − ∑_{τe=v} h′(e)) =

= ∑_{e edge in Γ} (h′(e) − h′(e)) = 0.

Thus Div_{v}(h′) = 0 for any v, which contradicts (3).

\underline{Case 2.} If Γ′ contains s vertex v′ from T and v′ ≠ ιe′, then do the same construction in the other direction: let Γ′′ be the union of all the paths (e_{1}, ..., e_{n}) in Γ_{+}(h) with e_{n} = e′. Now Γ′′ does not contain a vertex from T, except possibly for τe′, because otherwise we could connect such a vertex to v′ with a T -path in Γ_{+}(h), which would contradict the assumptions of the lemma. So the same argument as in Case 1 works for Γ′′.

Each T -path p in Γ(f) gives rise to an integer 1-chain in Γ which we also denote p. As before, 𝔸 denotes ℤ, ℚ, or ℝ.

Theorem 6. Let Γ be a graph, T be a set of vertices in Γ and f be a summable 1-chain on Γ with coefficients in 𝔸 and supp(∂f) ⊆ T.

(a) There is a countable family P = {p_{1}, p_{2}, ...} of simple T -paths in Γ_{+}(f) and a sequence {α_{i}} in [0, ∞) ∩ 𝔸 such that f = ∑_{i} α_{i}p_{i} and |f|_{1} = ∑_{i} α_{i}|p_{i}|_{1}.

(b) If the above f happens to have finite support, then P can be chosen to be finite.

This is a generalization of Theorem 4 and our proof of Theorem 6 follows the lines of the proof of Theorem 3.3 in [1] (cf also Lemma 3.6 in [4]), though we make the proof a bit more explicit without referring to the Zorn’s lemma. Proof of Theorem 6. Since f is summable, then Γ_{+}(f) has only countably many edges. Let P = {p_{1}, p_{2}, ...} be the (countable) set of all simple T -paths in Γ_{+}(f). If the support of f is finite, then P is finite.

For two summable 1-chains f′ and f′′ in Γ we will write f′ ⪯ f′′ if f′(e) ≤ f′′(e) for each edge in Γ(f) (with the orientation defined in Γ(f)).

Let α_{1} be the (non-negative) real number which is maximal among those satisfying

α_{1}p_{1} ⪯ f.

Note that α_{1} ∈ [0, ∞) ∩ 𝔸 since α_{1} coincides with the minimal value of f on the edges of p_{1}. Continue inductively: if α_{1}, ..., α_{i−1} are constructed, let α_{i} be the maximal real number satisfying

α_{1}p_{1} + ... + α_{i−1}p_{i−1} + α_{i}p_{i} ⪯ f.

Inductively we see that each α_{i} is in [0, ∞) ∩ 𝔸.

Denote f_{i} := α_{1}p_{1} + ... + α_{i}p_{i}. Since all the chains are non-negative,

|f_{i}|_{1} = α_{1}|p_{1}|_{1} + ... + α_{i}|p_{i}|_{1}.

The sequence f_{i} is monotone and bounded by the summable chain f, so the chain

f̄:= ∑_{i} α_{i}p_{i}

is well defined and satisfies

|f̄|_{1} := ∑_{i} α_{i}|p_{i}|_{1}.

Also, 0 ⪯ f̄ ⪯ f, therefore 0 ⪯ f − f̄ ⪯ f.

We prove now that Γ_{+}(f −f̄) does not contain nontrivial simple T -paths. If it contained a nontrivial simple T -path p, then αp ⪯ f − f̄ ⪯ f for some α > 0. Therefore p would be a simple path in Γ_{+}(f) as well, i.e. p = p_{i} for some i, and

f_{i} + αp_{i} ⪯ f̄+ αp_{i} ⪯ f.

This contradicts the maximality in the definition of α_{i}.

Also supp(∂(f − f̄)) ⊆ supp(∂f) ∪ supp(∂f̄) ⊆ T, so, by Lemma 5, f − f̄ = 0, i.e. f = f̄ = ∑_{i} α_{i}p_{i} and |f|_{1} = |f̄|_{1} = ∑_{i} α_{i}|p_{i}|_{1}. This finishes the proof of Theorem 6.

4. Linear isoperimetric inequalities.

One could mean many different things by “a linear homological isoperimetric inequality”. In this section we attempt presenting a comprehensive list of possible interpretations and show that they are all equivalent to hyperbolicity. Essentially all this was shown by S. M. Gersten (see [1, 3, 5, 7]), we just collect the statements in one theorem and sketch the proofs.

Theorem 7. Let G be a finitely presentable group G and X be a simply connected cellular 2-complex with a free cocompact G action. Then the following statements are equivalent.

(0) G is hyperbolic.

(1) There exists K_{1} ≥ 0 such that, for any b ∈ B_{1}(X, ℤ), |b|_{fc,ℤ} ≤ K_{1}|b|_{1}. (1’) There exists K′_{1} ≥ 0 such that for any b ∈ B_{1}(X, ℤ) there exists a ∈ C_{2}(X, ℤ) with

∂a = b and |a|_{1} ≤ K′_{1}|b|_{1}.

(2) There exists K_{2} ≥ 0 such that, for any b ∈ B_{1}(X, ℤ), |b|_{fc,ℚ} ≤ K_{2}|b|_{1}.

(3) There exists K_{3} ≥ 0 such that, for any b ∈ B_{1}(X, ℚ), |b|_{fc,ℚ} ≤ K_{3}|b|_{1}. (3’) There exists K′_{3} ≥ 0 such that for any b ∈ B_{1}(X, ℚ) there exists a ∈ C_{2}(X, ℚ) with

∂a = b and |a|_{1} ≤ K′_{3}|b|_{1}.

(4) There exists K_{4} ≥ 0 such that, for any b ∈ B_{1}(X, ℝ), |b|_{fc,ℝ} ≤ K_{4}|b|_{1}. (4’) There exists K′_{4} ≥ 0 such that for any b ∈ B_{1}(X, ℝ) there exists a ∈ C_{2}(X, ℝ) with

∂a = b and |a|_{1} ≤ K′_{4}|b|_{1}.

(5) There exists K_{5} ≥ 0 such that, for any b ∈ B_{1}(X, ℝ) ⊆ B_{1}^{(1)}(X, ℝ), |b|_{f1,ℝ} ≤ K_{5}|b|_{1}.

(6) There exists K_{6} ≥ 0 such that, for any b ∈ B_{1}^{(1)}(X, ℝ), |b|_{f1,ℝ} ≤ K_{6}|b|_{1}. (6’) There exists K′_{6} ≥ 0 such that, for any b ∈ B_{1}^{(1)}(X, ℝ), there exists a ∈ C_{2}^{(1)}(X, ℝ)

with ˆ∂_{2}a = b and |a|_{1} ≤ K′_{6}|b|_{1}.

(7) There exists K_{7} ≥ 0 such that, for any z ∈ Z_{1}^{(1)}(X, ℝ), |z|_{f1,ℝ} ≤ K_{7}|z|_{1}. (7’) There exists K′_{7} ≥ 0 such that, for any z ∈ Z_{1}^{(1)}(X, ℝ), there exists a ∈ C_{2}^{(1)}(X, ℝ)

with ∂a = z and |a|_{1} ≤ K′_{7}|z|_{1}.

Later in the paper we will only need implication (5) ⇒ (0), and this is the only implication we actually prove here. It also follows from Gersten’s results and we present a direct proof which in turn is based on ideas of [1, 3, 12]. Sketch of proof of Theorem 7. Equivalences (1) ⇔ (1′), (3) ⇔ (3′), (4) ⇔ (4′),

(6) ⇔ (6′), (7) ⇔ (7′) follow from the definition of the filling norm.

(0) ⇒ (1′) Hyperbolic groups are defined as the ones having a linear isoperimetric inequality for filling edge-loops with combinatorial disks. Each b ∈ B_{1}(X, ℤ) is a sum of such loops (viewed as chains). Let a be the sum of fillings for these loops. (See [6] where the converse was also proved.)

(1) ⇒ (2) is obvious because | · |_{fc,ℚ} ≤ | · |_{fc,ℤ} on B_{1}(X, ℤ). Similarly, (4) ⇒ (5).

(2) ⇒ (3) Let Γ := X^{(1)} and b ∈ B_{1}(X, ℚ). Since b is a summable cocycle, being the boundary of a summable 2-chain, then, by the above Theorem 4, b = ∑_{C} g(c)c for some g : C → [0, ∞). The main point here is that the elements of C are integer 1-cycles in X. Also, since g is rational, g can be chosen to take rational values (see Theorem 6 for the proof.) Fill each c ∈ C by an integer 2-chain a_{c} with a linear isoperimetric inequality, then ∑_{C} g(c)a_{c} is a rational filling of b with a linear isoperimetric inequality.

(3′) ⇒ (4′) and (5) ⇒ (7) are analogous to (2) ⇒ (3). (See also Proposition 3.6 in [1] for the proof of (1) ⇔ (3) ⇔ (4).)

(7′) ⇒ (6′), (6) ⇒ (5) are obvious.

We devote the rest of this section to the proof of the remaining implication

(5) ⇒ (0) It immediately follows from the following proposition (cf Proposition 5.4 in [3].)

Proposition 8. Let G and X be as in the hypotheses of Theorem 7 and suppose G is not hyperbolic. Then there exist C > 0 and a sequence of geodesic quadrilaterals w in X^{(1)} with

|w|_{1} → ∞  and  |w|_{f1,ℝ} ≥ C(|w|_{1})^{2}.

Here, abusing the notations, we identify the quadrilateral w and the corresponding integral 1-chain. Proof. We modify the techniques of [2, 3] which in turn used a modification of the Ol’shanskii’s method of layers [12].

Each 2-cell in X is a polygon whose sides are glued to edges in X^{(1)}. Let M be the maximal number of sides over all 2-cells in X. In particular, all combinatorial boundaries of 2-cells in X have diameter at most M, and also, for any 2-chain a in X,

|∂a|_{1} ≤ M|a|_{1}. (4)

If G is not hyperbolic, Ol’shanskii shows that there are arbitrarily thick geodesic quadrilaterals in X^{(1)}, and Gersten states more precisely that (see Proposition 5.2 in [3] in which we make the substitution t = 8Mr) there exist

• a sequence of integers r tending to infinity,

• a geodesic quadrilateral w = w(r) for each r,

• a (geodesic) side S in each w, and

• a subinterval [x, y] ⊆ S with the midpoint z such that

• the length of [x, y] is 4Mr,

• the distance from z to the sides of w other than S is ≥ 4Mr, and

• the perimeter of w is ≤ 80Mr. The last property implies that

|w|_{1} ≤ 80Mr. (5)

Since the interval [x, y] does not intersect other sides of the quadrilateral,

|w|_{1} ≥ 4Mr → ∞ as r → ∞.

For each positive integer k ≤ r, let B_{k} be the simplicial ball of radius 4Mk centered at z. If a is any summable chain with ∂a = w, denote a_{k} the restriction of a to (the 2-cells of) B_{k} \ B_{k−1} (defining by zero outside). Let s_{k} and t_{k} be the two points on [x, y] which are at distance 4Mk − 2M from z (see Fig. 1), and

A_{k} := {v ∈ [x, y] | 4M(k − 1) ≤ d(z, v) ≤ 4Mk − 2M}
A′_{k} := {v ∈ [x, y] | 4Mk − 2M ≤ d(z, v) ≤ 4Mk}.

∂a_{k} is a 1-cycle which takes value 1 on the edges of [x, y] incident on s_{k} and t_{k} (since these edges are “deep inside” B_{k} \ B_{k−1}). Also, ∂a_{k} splits as the sum of a 1-chain q_{k} supported in

{M − neighborhood of B_{k−1}} ∪ A_{k},

and a 1-chain q′_{k} supported in

\{M − neighborhood of \overline{X \ B_{k}}\} ∪ A′_{k}.

These two supports intersect only in T := {s_{k}, t_{k}}, and the above observation implies that

∂q_{k} = t_{k} − s_{k} and ∂q′_{k} = s_{k} − t_{k}. (6)

By Theorem 6, each q_{k} splits as a (finite) linear combination of T -paths

q_{k} = ∑_{i} α_{i}p_{i}

[figure: z; x; y; s_{k}; t_{k}; S; q′_{k}; q_{k}; B_{k−1}; w]

Figure 1. The filling a of b.

such that

α_{i} ≥ 0 and |q_{k}|_{1} = ∑_{i} α_{i}|p_{i}|_{1}.

Also, (6) says that ∑_{i} α_{i} ≥ 1. Those T -paths p_{i} which have non-trivial boundary must be of length at least d(s_{k}, t_{k}) ≥ 8Mk − 4M, so

|q_{k}|_{1} = ∑_{i} α_{i}|p_{i}|_{1} ≥ ∑_{i} α_{i}(8Mk − 4M) ≥ 8Mk − 4M.

The same holds for q′_{k}, hence

|∂a_{k}|_{1} = |q_{k}|_{1} + |q′_{k}|_{1} ≥ 2(8Mk − 4M) = 16Mk − 4M.

By (4),

|a_{k}|_{1} ≥ \frac{1}{M} |∂a_{k}|_{1} ≥ 16k − 4,

and since the chains a_{k} have disjoint supports,

|a|_{1} ≥ ∑_{k=1}^{r} |a_{k}|_{1} ≥ ∑_{k=1}^{r}(16k − 4) = 16\frac{r(r + 1)}{2} − 4r ≥ 8r^{2}.

Then, by (5),

|a|_{1} ≥ 8 (\frac{|w|_{1}}{80M} )^{2} = \frac{1}{800M^{2}} · (|w|_{1})^{2}.

Since this is true for any summable filling a of w, then

|w|_{f1,ℝ} ≥ \frac{1}{800M^{2}} · (|w|_{1})^{2},

so we put C := \frac{1}{800M^{2}} . Proposition 8 and Theorem 7 are proved.

5. The characterization.

In this section we characterize hyperbolic groups by bounded cohomology.

Let G be a finitely presentable group and M(G) be one of the following six classes of modules.

• M_{6}(G), the class of \underline{bounded ℚG-modules}. A ℚG-module is called \underline{bounded} if it is normed as a vector space over ℚ and G acts on it by linear operators of uniformly bounded norms.

• M_{5}(G), the class of \underline{isometric ℚG-modules}. A ℚG-module is called \underline{isometric} if it is bounded and, moreover, the G-action preserves its norm.

• M_{4}(G), analogously, the class of \underline{bounded ℝG-modules}.

• M_{3}(G), analogously, the class of \underline{isometric ℝG-modules}.

• M_{2}(G), analogously, the class of \underline{bounded Banach ℝG-modules}, i.e. bounded ℝGmodules which are Banach spaces with respect to their norms.

• M_{1}(G), analogously, the class of \underline{isometric Banach ℝG-modules}. Obviously, M_{1}(G) ⊆ M(G) ⊆ M_{6}(G) for each such M(G).

Theorem 3 promised in the introduction is stated more precisely as

Theorem 9. Let G be a finitely presentable group, and M(G) be one of the classes M_{1}(G) − M_{6}(G) described above. Then the following statements are equivalent.

(a) G is hyperbolic.

(b) The map H_{b}^{2}(G, V) → H^{2}(G, V) is surjective for any V ∈ M(G).

(c) The map H_{b}^{i}(G, V) → H^{i}(G, V) is surjective for any i ≥ 2 and any V ∈ M(G).

Proof. (a) ⇒ (c) was shown in [10] for the largest class M_{6}(G). Property (c) for M_{6}(G) implies the same property for each of the classes M_{1}(G) − M_{6}(G). The implication (c) ⇒ (b) is obvious. It only remains to prove (b) ⇒ (a) for the class M_{1}(G).

Since G is finitely presentable, there exists a contractible cell complex X with a free cellular G-action such that the induced action on the 2-skeleton of X is cocompact. We take V to be B_{1}^{(1)}(X, ℝ) with the filling norm | · |_{f1,ℝ} which we denote for simplicity by | · |_{f}. The vector space V is Banach as the quotient of the Banach space C_{2}^{(1)}(X, ℝ) by the (closed) kernel of ˆ∂_{2} : C_{2}^{(1)}(X, ℝ) → C_{1}^{(1)}(X, ℝ). In particular, V ∈ M_{1}(G).

Let Y be the geometric realization of the homogeneous bar-construction for G, i.e. Y is the simplicial complex whose k-simplices are labeled by ordered (k + 1)-tuples [x_{0}, ..., x_{k}] of elements of the group G, and each simplex labeled [x_{0}, ..., x̂_{i}, ..., x_{k}] is identified with the i-th face of [x_{0}, ..., x_{k}]. The action of G on Y is diagonal:

g · [x_{0}, ..., x_{k}] := [g · x_{0}, ..., g · x_{k}].

Let C_{∗}^{X} and C_{∗}^{Y} be the augmented chain complexes

... ⟶ C_{2}(X, ℝ) ⟶^{∂_{2}} C_{1}(X, ℝ) ⟶^{∂_{1}} C_{0}(X, ℝ) ⟶^{ϵ} ℝ ⟶ 0

and

... ⟶ C_{2}(Y, ℝ) ⟶^{∂_{2}} C_{1}(Y, ℝ) ⟶^{∂_{1}} C_{0}(Y, ℝ) ⟶^{ϵ} ℝ ⟶ 0 ,

respectively. Both X and Y are contractible, hence C_{∗}^{X} and C_{∗}^{Y} are acyclic. Both C_{∗}^{X} and C_{∗}^{Y} have free ℝG-modules in each non-negative dimension. Also C_{i}^{X} is finitely generated as an ℝG-module for i = 0, 1, 2.

By standard homological algebra, there exist homotopy equivalences φ_{∗} : C_{∗}^{Y} → C_{∗}^{X} and ψ_{∗} : C_{∗}^{X} → C_{∗}^{Y} which are identities in dimension −1. It is important that φ_{∗} and ψ_{∗} are chain maps in the category of ℝG-modules, in particular, for each i, φ_{i} and ψ_{i} are linear maps commuting with the G-action.

For the dual cochain complexes

C_{X}^{i} := C^{∗}(X, V) = Hom_{ℝG}(C_{i}^{Y} , V) and C_{Y}^{i} := C^{∗}(Y, V) = Hom_{ℝG}(C_{i}^{Y} , V)

and the dual maps

φ^{∗} : C_{X}^{∗} → C_{Y}^{∗} and ψ^{∗} : C_{Y}^{∗} → C_{X}^{∗},

the cochain map ψ^{∗} ∘ φ^{∗} is homotopic to the identity map, hence ψ^{∗} ∘ φ^{∗} induces the identity map on cohomology H^{∗}(G, V) in the positive dimensions.

The \underline{universal cocycle} in C_{X}^{2} is the 2-cochain u : C_{2}^{X} → V which coincides with the composition

C_{2}(X, ℝ) ⟶^{∂_{2}} B_{1}(X, ℝ) ↪ B_{1}^{(1)}(X, ℝ).

One checks that u is indeed a cocycle. By the above observations,

u = (ψ^{2} ∘ φ^{2})(u) + δv (7)

for some 1-cochain v : C_{1}^{X} → V.

Since φ^{2}(u) is a cocycle in C_{Y}^{2} and the map H_{b}^{2}(G, V) → H^{2}(G, V) is surjective by the assumption,

φ^{2}(u) = u′ + δv′, (8)

for a 1-cochain v′ ∈ C_{Y}^{1} and a bounded 2-cocycle u′ ∈ C_{Y}^{2} , i.e.

|u′|_{∞} < ∞. (9)

The above information is demonstrated by the diagrams

[figure: a, b ∈ C_{∗}^{X}; ψ_{∗}; =; C_{∗}(X, ℝ); and; u, v ∈ C_{X}^{∗}; φ^{∗}; =; C^{∗}(X, V); C_{∗}^{Y}; φ_{∗}; =; C_{∗}(Y, ℝ); u′, v′ ∈ C_{Y}^{∗}; ψ^{∗}; =; C^{∗}(Y, V).]

We also have the standard pairings ⟨·, ·⟩ : C_{X}^{i} ⊕ C_{i}^{X} → V and ⟨·, ·⟩ : C_{Y}^{i} ⊕ C_{i}^{Y} → V.

Fix any vertex y in Y (i.e. y ∈ G). For each 1-chain b = ∑_{y_{0},y_{1}∈G} β_{[y_{0},y_{1}]}[y_{0}, y_{1}] in Y, \underline{the cone} over b with vertex y is the 2-chain

[y, b] := ∑_{y_{0},y_{1}∈G} β_{[y_{0},y_{1}]}[y, y_{0}, y_{1}].

If b happens to be a cycle, then ∂[y, b] = b. Obviously,

|[y, b]|_{1} = |b|_{1}.

In other words, [y, b] is a filling of b in Y, and the above formula says that these fillings satisfy a linear isoperimetric inequality in Y. We want the same property in X to show hyperbolicity, so we will use the maps φ^{∗} and ψ^{∗} to “transfer” the linear isoperimetric inequality from Y to X using the fact that the universal cocycle is cohomologous to a bounded cocycle.

Note also that if α is a cocycle in C_{Y}^{i} and c ∈ C_{i}^{Y}, then c − [y, ∂c] is a cycle, hence a boundary, so ⟨α, c − [y, ∂c]⟩ = 0 and

⟨α, c⟩ = ⟨α, [y, ∂c]⟩. (10)

Now with the above setup we can finish the proof of Theorem 9. Pick any 1-boundary b ∈ B_{1}(X, ℝ) and any 2-chain a with ∂a = b. We will show that |b|_{f} ≤ K|b|_{1} for some uniform constant K, therefore G is hyperbolic by implication (5) ⇒ (0) in Theorem 7.

By equality (7),

b = ∂a = ⟨u, a⟩ = ⟨(ψ^{2} ∘ φ^{2})(u) + δv, a⟩ = ⟨(ψ^{2} ∘ φ^{2})(u), a⟩ + ⟨v, b⟩.

Since φ^{2}(u) is a cocycle, and using (10) and (8),

⟨(ψ^{2} ∘ φ^{2})(u), a⟩ = ⟨φ^{2}(u), ψ_{2}(a)⟩ = ⟨φ^{2}(u), [y, ∂(ψ_{2}(a))]⟩ =

= ⟨φ^{2}(u), [y, ψ_{1}(b)]⟩ = ⟨u′ + δv′, [y, ψ_{1}(b)]⟩ = ⟨u′, [y, ψ_{1}(b)]⟩ + ⟨v′, ∂[y, ψ_{1}(b)]⟩ =

= ⟨u′, [y, ψ_{1}(b)]⟩ + ⟨v′, ψ_{1}(b)⟩ = ⟨u′, [y, ψ_{1}(b)]⟩ + ⟨ψ^{1}(v′), b⟩.

So, combining the two formulas above,

b = ⟨u′, [y, ψ_{1}(b)]⟩ + ⟨ψ^{1}(v′) + v, b⟩,
|b|_{f} ≤ |⟨u′, [y, ψ_{1}(b)]⟩|_{f} + |⟨ψ^{1}(v′) + v, b⟩|_{f} ≤
≤ |u′|_{∞} · |[y, ψ_{1}(b)]|_{1} + |ψ^{1}(v′) + v|_{∞} · |b|_{1} =
= |u′|_{∞} · |ψ_{1}(b)|_{1} + |ψ^{1}(v′) + v|_{∞} · |b|_{1} ≤
≤ (|u′|_{∞} · |ψ_{1}|_{∞} + |ψ^{1}(v′) + v|_{∞}) · |b|_{1}.

This will give the desired linear isoperimetric inequality once we prove that all the norms in the parentheses are finite.

The cochain u′ is bounded by definition (by a constant depending only on the choice of G and X, see (9)). The maps ψ_{1} : C_{1}^{X} → C_{1}^{Y} and ψ^{1}(v′) + v : C_{1}^{X} → V are both linear maps commuting with the G-action. Their boundedness (by constants depending only on G and X) is immediate from the following simple observation which deserves being stated as a separate lemma.

Lemma 10. Let (W, | · |_{1}) and (W′, | · |) be two normed vector spaces over ℝ, where | · |_{1} is the ℓ_{1}-norm on W with respect to some basis. Suppose a group G acts

• on W permuting the basis so that there are only finitely many orbits of basis elements, and also

• on W′ preserving the norm | · |. If f : W → W′ is a linear map commuting with the G-action, then f is bounded, i.e. |f|_{∞} < ∞.

Proof. If w_{1} and w_{2} are two basis elements of W in the same G-orbit, i.e. w_{1} = g · w_{2}, then

|f(w_{1})| = |f(g · w_{2})| = |g · f(w_{2})| = |f(w_{2})|.

Since there are only finitely many G-orbits of basis elements in W, |f(·)| takes only finitely many values on the basis elements, hence |f|_{∞} < ∞. This proves Lemma 10 and Theorem 9.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

e-mail: mineyev@mpim-bonn.mpg.de, mineyev@mathstat.usouthal.edu

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http://www.math.utah.edu/~mineyev/math/
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