K-Theory 6: 235-265, 1992. © 1992 Kfuwer Academic Publishers. Printed in the Netherlands. 235 Assembly Maps, K-Theory, and Hyperbolic Groups C. OGLE Department of Mathematics, Ohio State University, Columbus, 0H43210, U.S.A. (Received: March 1992) Abstraet. Following Connes and Moscovici,we show that the Baum-Connes assembly map for K,(C~*n) is rationally injectivewhen n is word-hyperbolic,implying the Equivariant Novikov conjecture for such groups. Using this result in topological K-theory and BoreI-Karoubi regulators, we also show that the corresponding generalized assembly map in algebraic K-theory is rationally injective. Key words. Cyclic cohomology,elliptic group ¢ocycles, generalized assembly map, hyperbolic groups. Introduction The classical assembly map for K-theory arises in the following way (cf. [Lod]): given a commutative ring R, an algebra A over R with unit and a representa- tion p: F ~ GL(A), where F is a discrete group, there is a map of spectra ~BV+ A K(R)~K(A). (1) To define this map, note first that p determines a map of spaces Bfi:BF-~ K(A), where K(A) is the zeroth space of the spectrum K(A), hence upon passing to adjoints a map of spectra ~B~:__Z~BF+ --, K(A). (2) The algebra structure of A over R makes K(A) a module-spectrum over the ring- spectrum K(R). This means that there is a well-defined pairing K(A) /~ K(R) _~(A,R) , ~_K(A); (3) (1) is then the composition determined by (2) and (3) Z~BF+ A K(R) -Z-~BpAid-K(A) A K(R) __~A,R), ~K(A). (4) 236 c. OGLE The above description works in both the algebraic and topological cases. In the algebraic case, R is a discrete or simplicial commutative ring, A is usually R[F], and K(S) denotes the Quillen (or Waldhausen) K-theory spectrum of S. In the topological case, R is typically a commutative Banach algebra and A a topological algebra over R containing R[F]. For example, one could take R = C, and A = C*F a suitable C*-algebra completion of C[F], in which case the assembly map (4) would then give the usual operator algebra K-theory assembly map for C*F. In recent years, this map has been considerably generalized, one motivation being to account for the part of K-theory or Witt theory arising from torsion in F. The first such generalization of note was due to Quinn for algebraic K-theory ([Q1], [Q2]; see also [FJ]) who showed that for polycyctic groups the resulting map is an equivalence ([Q2]). In a similar vein, but completely independently, Baum and Connes [BC1] have defined an assembly map K,(Er) K,(C*r). (5) Their definition is essentially analytical. The group on the left is a certain direct limit of equivariant Kasparov KK-groups, where EF is the universal space for proper F actions. This group admits a Chern character which produces an isomorphism K,(EF) (~ C ~ ~ H,(BCg;C)~) (K,(C) ~) C). (6) z (a) c z ord(g) < o~ The sum on the right-hand side is over all conjugacy classes in F of finite order, where Cg = centralizer of g in F. Baum and Connes have conjectured that #(F) in (5) is an isomorphism for all discrete groups F. The aim of this paper is to compute the image of the rationalized Baum-Connes (BC) assembly map for hyperbolic groups. To do this, we begin with a homotopic reformulation of their map in Section 1 (the proof that these two descriptions agree rationally will appear in [BHO]). As a consequence of this homotopy-theoretic description, we are able to define analogues of the BC assembly map in both algebraic and Hermitian K-theory (hence, also Witt theory after inverting 2). Unlike the usual assembly map, this requires that the coefficient ring R satisfy certain properties. Let Sr=•[tp,eFfl 2~P/ [~geFwithord(g)<ooandp[ord(g) }3. Our generalized assembly map for algebraic K-theory is then a homomorphism (see (1.22)): A(r): @ U,(BC.;K_°"(Sr)) ® Sr K ,(sr[r:l) ® st. (7) (a) z z ord(0) < oo H,(X+;Kalg(sr)) as usual denotes the homology of X+ with coefficients in the algebraic K-theory spectrum Kalg(Sr). It is important in our case that we work with ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 237 the nonconnective K-theory spectrum of Sr, because of the way Ko(Sr) comes into play. Restricted to, or localized at the conjugacy class (1) this assembly map is just the usual one of (4) tensored (over Z) with St. The topological case works the same way. An interesting consequence of the above-mentioned [BHO] is that the usual BC assembly map (after complexification) factors as (~ H,(BCg;C ) @ K.(C)~ K~'(C[F])[fi -t] @ C <a> z or.(0) < (8) top --, K, (c r) ® c ;g where K~P(C[F])[f1-1] denotes the Bott-periodized topological K-theory (in the sense of Snaith and Thomason) of C[F], where C[F] is topologized by the fine topology. Having defined our map, we proceed in Section 2 to derive the explicit formulas in cyclic theory needed later on to detect the image of the assembly map. The basic result here is due to Burghelea [Bull who computed the cyclic homology and cohomology of R[F] for suitable R. Burghelea's results show that the left-hand side of (8) appear as a summand of the cyclic homology group HC.(C[F]). Our main result in this section is an explicit formula for the elliptic cyclic cocycle ~c,<g> e C"(R[F]) = cyclic n-cochains (over R) on R[F] determined by a normalized cocycle c e Cn(BCg; R). We now deal with the injectivity question, For a given discrete group F, the conjecture that KU,(BF) (~) Q --* K,(C*F) @ Q (9) z 2 is injective is due to Kasparov ([K1]) who has labelled it the (rationalized) Strong Novikov conjecture (SNC), and shown that injectivity of (9) implies the standard Novikov Conjecture on the homotopy-invariance of the higher Hirzebruch signa- tures of an even-dimensional closed, oriented manifold M 2" with rcl(M ~') = F. Kasparov [K1] has shown SNC to be true when F is a discrete subgroup of a Lie group G (rcoG finite); more recently, Connes and Moscovici [CM] have shown that SNC is true for finitely-generated word hyperbolic groups in the sense of Gromov [Gr]. Our main result, proved in Section 5 is THEOREM A (cf. Theorem 5.1). The assembly map At°P(F; C) @ C: (~) H,(BCo;K.(C)) @ C ~ K~°P(C*F) @ C (g) z z oN(o) < oo is injective if F is a finitely-generated word-hyperbolic group (in the sense of Gromov), Of course, localized at (1) this is the result of Connes and Moscovici. For hyperbolic groups with torsion, this extension of the Connes-Moscovici result is nontrivial. In order to prove it, we follow the approach of [CM], suitably adapted to 238 c. OGLE handle the elliptic conjugacy classes (g) # 1. There are two main difficulties in generalizing their approach. First, it may happen that the set of elements So of elements conjugate to g (So ~ F/Cg) is infinite. Second, the Jolissaint estimate ([Jol], p. 61) used in computing an upper bound for the norm of a cyclic cocycle derived from a (complex) group cocycle on BF does not work naturally in cyclic theory at the elliptic conjugacy classes (g)# (1). To deal with the first point, in Section 3 we introduce a modification of the Haagarup algebra, denoted H~,C,L(F). H~c,L(F) is contained in the Haagarup algebra. When F is word-hyperbolic, H~,c,z(F) is dense and holomor- phically closed in C*F. This type of rapid decay algebra was first studied by Harish- Chandra in the context of representation theory (I thank H. Moscovici for pointing this out to me). The main technical result used in the proof of Theorem A is THEOREM B (cf. Theorem 4.1). Let F be a finitely generated word-hyperbolic group, L a (hyperbolic) word-length function on F. Then there exists a constant C >1 1 such that for each elliptic class (g), integer n >~ 0, complex-valued cohomology class [~o] ~ H"(BCo; C) and (normalized) representative ~o of [~0], ze,(0> extends to a cyclic n-cocycle on H~,C,L(F) with values in l ~ where m = ord(g). The proof of Theorem B given in Section 4 uses a number of deep properties of hyperbolic groups, among them the result of Gersten and Short [GS] that the subgroups C o are hyperbolic, as well as the solution of the conjugacy problem for hyperbolic groups due to Gromov (see [Gr]). In the simplest case n = 0, [~o] = t, the theorem shows that the traces associated with an elliptic conjugacy class (g) extend over H~,c.L(F). In Section 5, we complete the proof of Theorem A by applying Theorem B to detect the image of our assembly map in K-theory under the Connes-Karoubi chern character. In Section 6, we axiomatize the properties F should satisfy in order that the proof of injectivity for At°p(F; C) apply. Among other classes, our axioms apply to finite Cartesian products of word-hyperbolic groups. Finally, in Section 7 we show that Karoubi's generalized Borel regulators (constructed in [Karl), together with the results of Section 5, imply the algebraic analogue of Theorem A.