Cyclic cohomology, the Novikov conjecture and hyperbolic groups

Abstract

(Received in revised form 27 April 1989) NOVIKOV'S conjecture on the homotopy invariance of higher signatures (28) can be formulated as follows: given a finitely presented group I and a compact oriented smooth manifold M, together with a continuous map +:M+BI, the generalized signatures (L(M)* $*(<), (Ml), where 5 runs over all classes in H*(BI, Q) and L(M) denotes the total Hirzebruch L-class of M, are homotopy invariants of the pair (M, $). In other words, if h: N+M is a hotiiotopy equivalence of oriented smooth manifolds, then (L(N)* h*($*(t)), (iV))=(L(M)*$*({), (Ml). The validity of this conjecture has been established, by a variety of techniques, for many groups I, most notably for closed discrete subgroups of finitely connected Lie groups. The latter result is due to Kasparov (24) and its proofs is based on bivariant K-theory. In this paper we present a new and more direct method for attacking the Novikov conjecture, which yields a proof of the conjecture for Gromov's (word) hyperbolic groups ( 183. These groups form an extremely rich and interesting class of finitely presented groups, which differs significantly, both in size and in nature, from the groups for which Novikov's conjecture was previously known. First of all, as pointed out by Gromov (18), they are generic among all finitely presented groups in the following sense: the ratio between the number of hyperbolic groups and all groups with a fixed number of generators and a fixed number of relations, each of length at most 1, tends to 1 when I+co (18,0.2(A)). Secondly, when adding at random relations to a (non-elementary) hyperbolic group, one obtains again a hyperbolic group (18,5.5). Thirdly, the cohomology of any finite polyhedron can be embedded into the cohomology of a hyperbolic group (18, 0.2(c)). Also, many of the hyperbolic groups exhibit exotic properties, like Kazhdan's property T ( 18, 5.61 or being non-linear (in a non-trivial way). Our approach is based on expressing the higher signatures in terms of the pairing between cyclic cohomology and K-theory (cf. (8)). The hyperbolicity assumption plays a twofold role: first, it ensures, via a deep result of Gromov (18; 8.3T-J, that every class < E Hk*' (BT, C) can be represented by a bounded group cocycle, and secondly, it enables us to make use of a critical norm estimate (first proved by Haagerup (20) for free groups), recently extended by Jolissaint (23) and de la Harpe (21) to hyperbolic groups. The paper is organized as follows. Using the Alexander-Spanier realization of the cohomology of a smooth manifold, reviewed in $1, we define in $2 localized analytic indices