BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS


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1 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS by IGOR MINEYEV (University of South Alabama, Department of Mathematics & Statistics, ILB 35, Mobile, AL 36688, USA) [Received June 00] Abstract A finitely presentable group G is hyperbolic if and only if the map H b (G, V ) H (G, V ) is surjective for any bounded Gmodule. The only if direction is known and here we prove the if direction. We also consider several ways to define a linear homological isoperimetric inequality.. Introduction The question of cohomological description of hyperbolicity was considered by S. M. Gersten who proved the following theorem. THEOREM ([3]). The finitely presented group G is hyperbolic if and only if H ( ) (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, ) complex with finitely many cells in the dimensions up to n. This theorem was generalized by the author [0] to higher dimensions: if G is hyperbolic then H( ) n (G, V ) = 0 for any n 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 definition in the next section). B. E. Johnson [8, Theorem.5] characterized amenable groups by the vanishing of H (L (G), X ), the first cohomology of the Banach algebra L (G). (The vanishing in higher dimensions also follows from his argument. Bounded cohomology is an example of the cohomology above.) In [, p. 068] G. A. Noskov also characterized amenable groups by the vanishing of the bounded cohomology for the positive dimensions. We present this result in the following form. THEOREM (Johnson [8]). For a group G the following statements are equivalent. (a) G is amenable. (b) H b (G, V ) = 0 for any bounded Gmodule V. (c) H i b (G, V ) = 0 for any i and any bounded Gmodule V. mineyev/math/ Quart. J. Math. 53 (00), c Oxford University Press 00
2 60 I. MINEYEV The expression bounded Gmodule V here is what we call a bounded Banach RGmodule, and V is the space conjugate to V. The main result of this paper is in a sense an analogue of the above two theorems: we characterize hyperbolic groups by bounded cohomology. It was shown in [9] that if G is a hyperbolic group then the map H i b (G, V ) H i (G, V ), induced by inclusion, is surjective for any bounded QGmodule V and any i (see the definitions in the next section). In the present paper we show the converse. When G is finitely presentable, the surjectivity of the above maps only in dimension implies hyperbolicity. Namely, we prove the following. THEOREM 3 For a finitely presentable group G, the following statements are equivalent. (a) G is hyperbolic. (b) The map H b (G, V ) H (G, V ) is surjective for any bounded Gmodule V. (c) The map H i b (G, V ) H i (G, V ) is surjective for any i and any bounded Gmodule V. Here by a bounded Gmodule we mean any one of the ten concepts M (G) to M 0 (G) described in section 5, see Theorem 9 for a more precise statement. It is quite interesting that the same property can be characterized by the l cohomology and the bounded cohomology, 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 further investigation. There are two crucial points in our proof. First, for finitely presentable groups hyperbolicity is equivalent to the existence of a linear isoperimetric inequality for real cycles. This equivalence was proved by Gersten. (It follows, for example, from [3, Proposition 3.6, Theorem 5., and Theorem 5.7] (stated above).) 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 cycles with summable chains implies hyperbolicity. Secondly, one needs to pick appropriate coefficients V. We take V to be the space of all boundaries of summable chains in a cell complex with a nice Gaction. Similar coefficients (but Zmodules rather than Rmodules or Cmodules), and the universal cocycle associated with them were used by S. M. Gersten [7, Chapter ] to introduce Zmetabolic (or simply metabolic) groups.. Preliminaries Let F stand for one of the fields Q, R or C, and let A stand for one of the rings Z, Q, R or C... Abelian group norms A normed abelian group A is an abelian group with an abelian group norm : A R + satisfying a =0 if and only if a = 0, and a + a a + a for all a, a A.
3 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS 6.. Norms A normed vector space W over F is a vector space with a norm :W R + satisfying w =0 if and only if 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..3. l norm Let a free Amodule M have a preferred basis {m i, i I }. The l norm on M (with respect to this basis) is given by α i m i := i I α i. i I The l norm is an abelian group norm. Moreover, if M happens to be a vector space over F, then it is a norm..4. l norm Suppose that (W, ) and (W, ) are normed vector spaces, where W is equipped with the l norm with respect to some preferred basis {w i, i I }. For a linear map ϕ : W W, the l norm of ϕ, ϕ, is the operator norm of ϕ, that is, ϕ is the smallest number K (possibly infinity) such that ϕ(w) K w for each w W. One checks that ϕ = sup ϕ(wi ). i I.5. Cell complexes By a cell complex we mean a combinatorial cell complex, that is, 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 cell can be viewed as a polygon. We always put the path metric, which is induced by assigning length to each edge, on the  skeleta of cell complexes. A cellular ball of radius r centred at a vertex x is the union of all closed cells whose vertices lie rclose to x. If X is a cell complex and C i (X, A) is the space of cellular ichains, we always give C i (X, A) the l norm with respect to the standard basis consisting of icells..6. Filling norm for compactly supported chains Gersten introduced this useful concept (sometimes also called standard norm ). If G is a finitely presentable group, then there exists a contractible cell complex X with a free cellular Gaction such that the induced action on the skeleton of X is cocompact. In particular, the boundary homomorphism : C (X, A) C (X, A) is bounded (with respect to the l norms in the domain and the target).
4 6 I. MINEYEV The filling norm fc,a on the space B (X, A) of boundaries is given by b fc,a := inf { a a C (X, A) and a = b }. In other words, fc,a is the abelian group norm induced by the map : C (X, A) C (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 if and only if w = 0 holds because fc,a dominates the l norm on B (X, A).).7. l completion For X as above and i =,, let C () i (X, F) be the set of all (absolutely) summable ichains in X. Since the Gaction on the skeleton of X is cocompact, the boundary homomorphism ˆ i : C () i (X, F) C () i (X, F), i =,, is well defined and bounded with respect to the l norm in the domain and in the target. Also, ˆ i commutes with the Gaction. Write B () i (X, F) := Im ˆ i+, Z () i (X, F) := Ker ˆ i..8. Filling norm for summable chains Now we consider the complete version of the filling norm. The norm f,f on the space B () (X, F) is given by b f,f := inf { a () a C (X, F) and a = b}. Again, since ˆ : C () (X, F) C() (X, F) is bounded, f,f dominates on B () (X, F) and therefore f,f is indeed a norm..9. Bounded cohomology An FGmodule is called bounded if it is normed as a vector space over F and G acts on it by linear operators of uniformly bounded norms. For a bounded FGmodule V, the bounded cohomology of G with coefficients in V, Hb (G, V ), is the homology of the cochain complex 0 Cb 0 (G, V ) δ 0 Cb (G, V ) δ Cb (G, V ) δ..., where Cb i (G, V ) := {α : Gi+ V α is a bounded Gmap} () and the coboundary map δ i is defined by δ i α ( [x 0,...,x i+ ] ) i+ := ( ) k α ( [x 0,..., x k,...,x i+ ] ). () k=0 Here G i+ is considered with the diagonal Gaction by left multiplication, and by bounded Gmap we mean a Gmap whose image is bounded with respect to the norm on V. Equivalently, C i b (G, V ) Hom FG(C i (G, F), V ) is the subspace of all FGmorphisms C i (G, F) V which are bounded as linear maps, where C i (G, F) is the space of all chains (that is, finite support functions) G i+ F given the l norm with respect to the standard basis G i+. The coboundary homomorphism δ in C i b (G, V ) is the dual of the boundary morphism in C i(g, F). The spaces C i b (G, V ) are naturally given the l norm dual to the l norm on C i (G, F).
5 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS Summable chains In this section we prove that certain summable 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. THEOREM 4 (Allcock Gersten []). If Ɣ is a graph and f is a summable realvalued cycle on Ɣ, then there is a countable coherent family C of simple circuits in Ɣ and a function g : C [0, ) such that f = C g(c)c. Coherent here means that, for any such f, f = C g(c) c. Let Ɣ be a graph. As a part of the structure, we assign an orientation to each edge of Ɣ. Therefore a chain on Ɣ is the same thing as a function on the edges of Ɣ. The orientation on Ɣ determines the initial vertex ιe and the terminal vertex τe for each edge e. Adirected path p in Ɣ is a sequence of edges (e,...,e n ) with τe i = ιe i+. The initial vertex ιp of the path p is ιe and the terminal vertex τ p of p is τe n. A directed path (e,...,e n ) is simple if the vertices ιe,...,ιe n are all distinct. If T is a set of vertices in Ɣ, we say that p is a T path if it is a directed path in Ɣ such that ιp,τp T or ιp = τ p (in other words, the homological boundary of p is supported in T ). Let f be a(n absolutely) summable chain in Ɣ. For a vertex v in Ɣ, define Div v ( f ) := f (e) f (e), ιe=v τe=v where e stands for edges in Ɣ (with the fixed orientation). By Ɣ( f ) we denote the same graph Ɣ but with an orientation on the edges chosen so that f (e) 0 for each edge e. Let Ɣ + ( f ) be the minimal subgraph of Ɣ( f ) containing all the edges e with f (e) = 0 (that is, f (e) >0). LEMMA 5 Let Ɣ be a graph and T be a set of vertices in Ɣ. If h is a summable real chain such that supp( h) T and Ɣ + (h) contains no nontrivial simple T paths, then h = 0. Proof. (cf. [, Lemma 3.]). Suppose to the contrary that h = 0, that is, there is an edge e in Ɣ + (h) (hence h(e )>0). Let Ɣ be the (nonempty) union of all directed paths (e,...,e n ) in Ɣ + (h) with e = e. Then e is the only edge in Ɣ incident on ιe (otherwise there would be a nontrivial directed edge cycle in Ɣ Ɣ + (h)). Case. We assume for the moment that Ɣ does not contain vertices from T, except possibly for ιe. Let h be the summable chain in Ɣ which coincides with h on Ɣ and takes the value 0 outside. If v is any vertex in Ɣ different from ιe, then v T supp( f ) and also, by definition, Ɣ contains all the edges e of Ɣ + (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 vvertex in Ɣ = eedgeinɣ Div v (h ) = vvertex in Ɣ ( h (e) h (e) ) = 0. ( h (e) h (e) ιe=v τe=v )
6 64 I. MINEYEV Thus Div v (h ) = 0 for any v, which contradicts (3). Case. If Ɣ contains s vertex v from T and v = ιe, then do the same construction in the other direction: let Ɣ be the union of all the paths (e,...,e n ) in Ɣ + (h) with e n = e.nowɣ 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 Ɣ + (h), which would contradict the assumptions of the lemma. So the same argument as in case works for Ɣ. Each T path p in Ɣ( f ) gives rise to an integer chain in Ɣ which we also denote by p. In the following theorem, A denotes Z, Q,orR. THEOREM 6 Let Ɣ be a graph, T a set of vertices in Ɣ and f a summable chain on Ɣ with coefficients in A and supp( f ) T. (a) There is a countable family P ={p, p,...} of simple T paths in Ɣ + ( f ) and a sequence {α i } in [0, ) A such that f = i α i p i and f = i α i p i. (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 [, proof of Theorem 3.3] (cf. also [4, Lemma 3.6]), though we make the proof a bit more explicit without referring to the Zorn s lemma. Proof. Since f is summable, Ɣ + ( f ) has only countably many edges. Let P ={p, p,...} be the (countable) set of all simple T paths in Ɣ + ( f ). If the support of f is finite, then P is finite. For two summable chains f and f in Ɣ we will write f f if f (e) f (e) for each edge in Ɣ( f ) (with the orientation defined in Ɣ( f )). Let α be the (nonnegative) real number which is maximal among those satisfying α p f. Note that α [0, ) A since α coincides with the minimal value of f on the edges of p. Continue inductively: if α,...,α i are constructed, let α i be the maximal real number satisfying α p + +α i p i + α i p i f. Inductively we see that each α i is in [0, ) A. Write f i := α p + +α i p i. Since all the chains are nonnegative, f i = α p + +α i p i. The sequence f i is monotone and bounded by the summable chain f, so the chain f := i α i p i is well defined and satisfies f := i α i p i. Also, 0 f f, therefore 0 f f f.
7 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS 65 We prove now that Ɣ + ( f f ) does not contain nontrivial simple T paths. If it contained a nontrivial simple T path p, then αp f f f for some α>0. Therefore p would be a simple path in Ɣ + ( f ) as well, that is, p = p i for some i, and f i + αp i f + αp i f. This contradicts the maximality in the definition of α i. Also supp( ( f f )) supp( f ) supp( f ) T so, by Lemma 5, f f = 0, that is, f = f = i α i p i and f = f = i α i p i. This finishes the proof of Theorem Linear isoperimetric inequalities One could mean many different things by a linear homological isoperimetric inequality. In this section we attempt to present a comprehensive list of possible interpretations and show that they are all equivalent to hyperbolicity. Most of this was shown by Gersten in various papers [, 3, 5, 7]; we just collect the statements into one theorem and sketch the proofs. THEOREM 7 Let G be a finitely presentable group G and let X be a simply connected cellular complex with a free cocompact G action. Then the following statements are equivalent. (0) G is hyperbolic. () There exists K 0 such that, for any b B (X, Z), b fc,z K b. ( ) There exists K 0 such that for any b B (X, Z) there exists a C (X, Z) with a = b and a K b. () There exists K 0 such that, for any b B (X, Z), b fc,q K b. (3) There exists K 3 0 such that, for any b B (X, Q), b fc,q K 3 b. (3 ) There exists K 3 0 such that for any b B (X, Q) there exists a C (X, Q) with a = b and a K 3 b. (4) There exists K 4 0 such that, for any b B (X, R), b fc,r K 4 b. (4 ) There exists K 4 0 such that for any b B (X, R) there exists a C (X, R) with a = b and a K 4 b. (5) There exists K 5 0 such that, for any b B (X, R) B () (X, R), b f,r K 5 b. (6) There exists K 6 0 such that, for any b B () (X, R), b f,r K 6 b. (6 ) There exists K 6 0 such that, for any b B() (X, R) there exists a C() (X, R) with ˆ a = b and a K 6 b. (7) There exists K 7 0 such that, for any z Z () (X, R), z f,r K 7 z. (7 ) There exists K 7 () 0 such that for any z Z (X, R) there exists a C() (X, R) with a = z and a K 7 z.
8 66 I. MINEYEV (8) There exists K 8 0 such that, for any b B (X, C), b fc,c K 8 b. (8 ) There exists K 8 0 such that for any b B (X, C) there exists a C (X, C) with a = b and a K 8 b. (9) There exists K 9 0 such that, for any b B (X, C) B () (X, C), b f,c K 9 b. (0) There exists K 0 0 such that, for any b B () (X, C), b f,c K 0 b. (0 ) There exists K 0 0 such that for any b B() (X, C) there exists a C() (X, C) with ˆ a = b and a K 0 b. () There exists K 0 such that, for any z Z () (X, C), z f,c K z. ( ) There exists K () 0 such that for any z Z (X, C) there exists a C() (X, C) with a = z and a K z. Remark. Later in the paper only implications (5) (0) and (9) (0) will be used. The author s contribution here is implication (5) (0) and the implications involving complex numbers. This article seems to be the first place where complex numbers are used for isoperimetric inequalities. Sketch of proof. Equivalences () ( ), (3) (3 ), (4) (4 ), (6) (6 ), (7) (7 ) (8) (8 ), (0) (0 ), () ( ) follow from the definition of the filling norm. (In each case, K can be taken to be K +.) (0) ( ) Hyperbolic groups are defined as those having a linear isoperimetric inequality for filling edgeloops with combinatorial disks. Each b B (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 is proved.) () () is obvious because fc,q fc,z on B (X, Z). Similarly, (4) (5). () (3) Let Ɣ := X () and b B (X, Q). Since b is a summable cocycle, being the boundary of a summable chain, it follows b: that Theorem 4 b = C g(c)c for some g : C [0, ). The main point here is that the elements of C are integer 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 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 () (3). (See also, [, Proposition 3.6] for the proof of () (3) (4).) (7 ) (6 ), (6) (5) are obvious. (4 ) (8 ) We need to pass from R to C. Ifb B (X, C), then Re b and Im b lie in b B (X, R), so, by (4 ), there exist a, a C (X, R) with a = Re b, a = Im b, a K 4 Re b, a K 4 Im b. Then (a + ia ) = b and a + a K 4 Re b + K 4 Im b K 4 b, so we set K 8 := K 4. (8 ) (4 ) We pass from C to R. Each b B (X, R) can be viewed as an element of B (X, C), therefore, by (8 ), there exists a C (X, C) with with a = b and a K 8 b. Let ā be the complex conjugate of a. Since b is real, (Re a) = Re b = b and Re a a K 8 b,sowe set K 4 := K 8.
9 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS 67 (5) (9), (6 ) (0 ), (7 ) ( ) These implications are similar to (4 ) (8 ). We devote the rest of this section to the proof of the remaining implication (5) (0). It follows immediately from the following proposition (cf. [3, Proposition 5.4]) 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 () with w and w f,r C( w ). Here, abusing notation, we identify the quadrilateral w and the corresponding integral chain. Proof. We modify the techniques of [], [3] which in turn used a modification of Ol shanskii s method of layers []. Each cell in X is a polygon whose sides are glued to edges in X (). Let M be the maximal number of sides over all cells in X. In particular, all combinatorial boundaries of cells in X have diameter at most M, and also, for any chain a in X, a M a. (4) If G is not hyperbolic, Ol shanskii shows that there are arbitrarily thick geodesic quadrilaterals in X (), and Gersten states more precisely that (see [3, Proposition 5.] 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 at least 4Mr, and the perimeter of w is at most 80Mr. This last property implies that w 80Mr. (5) Since the interval [x, y] does not intersect other sides of the quadrilateral, w 4Mr as r. For each positive integer k r, let B k be the simplicial ball of radius 4Mk centred at z. Ifa is any summable chain with a = w, denote by a k the restriction of a to (the cells of) B k \ B k (with a k = 0 outside). Let s k and t k be the two points on [x, y] which are at distance 4Mk M from z (see Fig. ), and A k := {v [x, y] 4M(k ) d(z,v) 4Mk M}, A k := {v [x, y] 4Mk M d(z,v) 4Mk}.
10 68 I. MINEYEV w q k q k B k x s k S z tk y Fig. The filling a of b. Here a k is a cycle which takes the value on the edges of [x, y] incident on s k and t k (since these edges are deep inside B k \ B k ). Also, a k splits as the sum of a chain q k supported in and a chain q k supported in {M neighbourhood of B k } A k, {M neighbourhood of X \ B k } A k. These two supports intersect only in T := {s k, t k : 0 < k r}, 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 such that α i 0 and q k = i α i p i. Those T paths p j which have nontrivial boundary must be of length at least d(s k, t k ) 8Mk 4M, and (6) says that j α j, so q k = i α i p i j α j (8Mk 4M) 8Mk 4M. The same holds for q k, hence a k = q k + q k (8Mk 4M) = 6Mk 4M.
11 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS 69 By (4), and since the chains a k have disjoint supports, a r a k k= a k M a k 6k 4, r r(r + ) (6k 4) = 6 Then, by (5), ( ) w a 8 = 80M 800M ( w ). Since this is true for any summable filling a of w, it follows that k= w f,r 800M ( w ), so we put C :=. Proposition 8 and Theorem 7 are proved. 800 M 4r 8r. 5. The characterization In this section we characterize hyperbolic groups by bounded cohomology. Let G be a finitely presentable group and let M(G) be one of the following ten classes of modules. M (G), the class of bounded QGmodules. AQGmodule is called bounded if it is normed as a vector space over Q and G acts on it by linear operators of uniformly bounded norms. M (G), the class of isometric QGmodules. AQGmodule is called isometric if it is bounded and, moreover, the Gaction preserves its norm. M 3 (G), analogously, the class of bounded RGmodules. M 4 (G), analogously, the class of isometric RGmodules. M 5 (G), analogously, the class of bounded Banach RGmodules, that is, bounded RGmodules which are Banach spaces with respect to their norms. M 6 (G), analogously, the class of isometric Banach RGmodules. M 7 (G), the class of bounded CGmodules. M 8 (G), the class of isometric CGmodules. M 9 (G), analogously, the class of bounded Banach CGmodules. M 0 (G), the class of isometric Banach CGmodules. Obviously, M 0 (G) M(G) M (G) for each such M(G). As promised in the introduction, Theorem 3 is stated more precisely as follows. THEOREM 9 Let G be a finitely presentable group, and M(G) one of the classes M (G) to M 0 (G) described above. Then the following statements are equivalent. (a) G is hyperbolic.
12 70 I. MINEYEV (b) The map Hb (G, V ) H (G, V ) is surjective for any V M(G). (c) The map Hb i(g, V ) H i (G, V ) is surjective for any i and any V M(G). Proof. (a) (c) was shown in [9] for the largest class M (G). Property (c) for M (G) implies the same property for each of the classes M (G) to M 0 (G). The implication (c) (b) is obvious. It only remains to prove (b) (a) for the class M 0 (G). Below we present the proof of (b) (a) for class M 6 (G), since real coefficients are often used. To show (b) (a) for class M 0 (G) the reader would just need to replace R by C everywhere. 5.. Proof of (b) (a) for class M 6 (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 skeleton of X is cocompact. We take V to be B () (X, R) with the filling norm f,r which we denote for simplicity by f. The vector space V is Banach as the quotient of the Banach space C () (X, R) by the (closed) kernel of ˆ : C () (X, R) C () (X, R). In particular, V M 6(G). Let Y be the geometric realization of the homogeneous barconstruction for G, that is, Y is the simplicial complex whose ksimplices are labelled by ordered (k + )tuples [x 0,...,x k ] of elements of the group G, and each simplex labeled [x 0,..., x i,...,x k ] is identified with the ith 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 CY be the augmented chain complexes... C (X, R) C (X, R) C 0 (X, R) ɛ R 0 and ɛ... C (Y, R) C (Y, R) C 0 (Y, R) R 0, respectively. Both X and Y are contractible, hence C X and CY are acyclic. Both C X and CY have free RGmodules in each nonnegative dimension. Also Ci X is finitely generated as an RGmodule for i = 0,,. Obviously, C X and CY coincide in the negative dimensions. By the standard uniqueness of resolutions property, there exist homotopy equivalences ϕ : C Y C X and ψ : C X C Y which are identities in dimension. It is important that ϕ and ψ are chain maps in the category of RGmodules, in particular, for each i, ϕ i and ψ i are linear maps commuting with the Gaction. For the dual cochain complexes C i X := C (X, V ) = Hom RG (C Y i, V ) and C i Y := C (Y, V ) = Hom RG (C Y i, V ) and the dual maps ϕ : C X C Y and ψ : CY 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 universal cocycle in C X is the cochain u : C X V which coincides with the composition C (X, R) B (X, R) B () (X, R).
13 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS 7 One checks that u is indeed a cocycle. By the above observations, u = (ψ ϕ )(u) + δv (7) for some cochain v : C X V. Since ϕ (u) is a cocycle in CY assumption, and the map H b (G, V ) H (G, V ) is surjective by the ϕ (u) = u + δv (8) for a cochain v CY and a bounded cocycle u CY, that is, u <. (9) The above information is demonstrated by the diagrams a, b C X ϕ C Y ψ = C (X, R) and u,v C X ϕ ψ = C (Y, R) u,v C Y = C (X, V ) = C (Y, V ). We also have the standard pairings, : C i X C i X V and, : CY i CY i V. Fix any vertex y in Y (that is, y G). For each chain b = y 0,y G β [y 0,y ][y 0, y ] in Y, the cone over b with vertex y is the chain [y, b] := β [y0,y ][y, y 0, y ]. y 0,y G If b happens to be a cycle, then [y, b] =b. Obviously, [y, b] = b. 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 i Y and c CY i, then c [y, c] is a cycle, hence a boundary, so α, c [y, c] = 0 and α, c = α, [y, c]. (0) Now with the above setup we can finish the proof of Theorem 9. Pick any boundary b B (X, R) and any chain a with a = b. We will show that b f K b for some uniform constant K, therefore G will be hyperbolic by implication (5) (0) in Theorem 7. (By implication (9) (0) in case of complex coefficients.) By equality (7), b = a = u, a = (ψ ϕ )(u) + δv,a = (ψ ϕ )(u), a + v, b.
14 7 I. MINEYEV Since ϕ (u) is a cocycle, and using (0) and (8), (ψ ϕ )(u), a = ϕ (u), ψ (a) = ϕ (u), [y, (ψ (a))] = ϕ (u), [y,ψ (b)] = u + δv, [y,ψ (b)] = u, [y,ψ (b)] + v, [y,ψ (b)] = u, [y,ψ (b)] + v,ψ (b) = u, [y,ψ (b)] + ψ (v ), b. So, combining the above two formulae, b = u, [y,ψ (b)] + ψ (v ) + v, b, b f u, [y,ψ (b)] + ψ (v ) + v, b f f u [y,ψ (b)] + ψ (v ) + v b = u ψ (b) + ψ (v ) + v b ( ) u ψ + ψ (v ) + v b. 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 : C X CY and ψ (v ) + v : C X V are both linear maps commuting with the Gaction. Their boundedness (by constants depending only on G and X) is immediate from the following simple observation which deserves the status of a lemma. LEMMA 0 Let (W, ) and (W, ) be two normed vector spaces over R, where is the l norm on W with respect to some basis. Suppose a group G acts on both W and W such that on W it permutes the basis so that there are only finitely many orbits of basis elements, and on W it preserves the norm. Then, if f : W W is a linear map commuting with the Gaction, then f is bounded, that is, f <. Proof. If w and w are two basis elements of W in the same Gorbit, that is, w = g w, then f (w ) = f (g w ) = g f (w ) = f (w ). Since there are only finitely many Gorbits of basis elements in W, f ( ) takes only finitely many values on the basis elements, hence f <. This proves Lemma 0 and Theorem 9. Acknowledgements The author thanks Steve Gersten and Vladimir Troitsky for helpful discussions, and very much appreciates the hospitality of the University of Melbourne and the MaxPlanckInstitut für Mathematik in Bonn in
15 BOUNDED COHOMOLOGY CHARACTERIZES HYPERBOLIC GROUPS 73 References. D. J. Allcock and S. M. Gersten, A homological characterization of hyperbolic groups, Invent. Math. 35 (999), J. Fletcher, Ph.D. Thesis, University of Utah, S. M. Gersten, A Cohomological Characterization of Hyperbolic Groups, Preprint available at gersten. 4. S. M. Gersten, Distortion and l homology, Proceedings of the 4th International Conference on the Theory of Groups held at Pusan National University, Pusan, August 0 6, (Eds Y. G. Baik, D. L. Johnson, and A. C. Kim), Gruyter, New York, (998), S. M. Gersten, A Note on Cohomological Vanishing and the Linear Isoperimetric Inequality, Preprint available at gersten. 6. S. M. Gersten, Subgroups of word hyperbolic groups in dimension, J. London Math. Soc. 54 (996), S. M. Gersten, Cohomological lower bounds for isoperimetric functions on groups, Topology 37 (998), B. E. Johnson, Cohomology in Banach algebras, Memoir, 7, American Mathematical Society, Providence, I. Mineyev, Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal. (00), I. Mineyev, Higher dimensional isoperimetric inequalities in hyperbolic groups, Math. Z. 33 (000), G. A. Noskov, Bounded cohomology of discrete groups with coefficients, Leningrad Math. J. (99), A. Y. Ol shanskii, Hyperbolicity of groups with subquadratic isoperimetric inequality, Internat. J. Algebra Comput. (99), 8 89.
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