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 injective when 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
~ B V + A K ( R ) ~ K ( A ) . (1)
To define this map, note first that p determines a map of spaces Bf i :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 ~ B F + 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
o r d ( 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
F f l 2~P }3 S r = • [ t p , e / [ ~ g e F w i t h o r d ( g ) < o o a n d p [ o r d ( g ) .
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
o r d ( 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])[f i - 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 S o of elements conjugate to g (S o ~ 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. This may be stated as
THEOREM C (cf. Theorem 7.1). Let R be the ring of integers of a number field
containing St . Then the assembly map
A~(F; C) @ C: (~ H,(BCa; K~lg(R)) @ C ~ K?~g(R[F]) @ C
ord(0) < go
is injective if F is a finitely-generated word-hyperbolic group (in the sense of Gromov).
A similar injectivity statement for the complexified Witt groups W,(Sr[F]) ®~ C also follows from Theorem A (cf. Cor. 5.5).
There is an interesting consequence of Theorem A to the geometry of manifolds (this application represents joint work with P. Baum and M. Davis). As Kasparov originally showed in [K1], the reason SNC implies NC is that under the assembly map (5), the topological index of the signature operator D~ of M 2n, viewed as an
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 239
element of the left-hand side of (5), maps to the image of the Michshenko symmetric signature o(D~) ~ W,(2~[F]) under the map
~;(Z[F]) ~ w,(c*r) - ~ K,(C*r).
a(D~) is, by the construction of the Michschenko Witt group, a homotopy invariant. Injectivity of the rationalized assembly map therefore implies the same for the topological index. This index may be viewed as the equivariant index of the signature operator D~ lifted to the universal cover ~r 2~ of M 2", on which F acts freely. Now as observed by Baum and Connes in [BC] and independently by M. Davis, one can associate to a proper action of F an equivariant index lying in the left-hand side of (7). Precisely, suppose one is given an oriented manifold X 2'~ equipped with an orient- ation-preserving proper action of F where X/F is compact. Let Indt(Dx,r) denote the topological index of the equivariant signature operator on X 2", viewed as n element of
@ H,(BCa;C) @ K,(C). tg}
ord(a) < co
EQUIVARIANT NOVIKOV CONJECTURE (ENC). Let X 2", y2n be two oriented manifolds, with F acting on both X 2~ and y2, in a proper and orientation-preserving way, with compact quotients. I f f : X 2" --* y2n is a F-equivariant map which is a rational homology equivalence, then
Indt(Dx,r) = Indt(Dr, r).
This conjecture is investigated in [BDO], where it is shown that rational injectivity of the BC assembly map for F implies ENC for that group (in this sense, rational injectivity of #(F) in (5) should be thought of as a strong ENC, following Kasparov). Therefore, as a consequence of Theorem A, [BHO] and [BDO], we conclude
THEOREM D. ENC is true for finitely generated word-hyperbolic groups F.
In fact, the equivariant Novikov conjecture is one of the main motivations for proving rational injectivity of the topological assembly map At°p(F; C) (or/2(F)).
I would like to thank P. Baum, A. Connes, H. Moscovici and N. Higson for useful conversations pertaining to this paper. It should be noted that subsequent to the circulation of a prior version of these results in March of 1991, N. Higson has announced the integral injectivity of the BC assembly map for word-hyperbolic groups (as well as cartesian products of such). S. Ferry and S. Weinberger have announced similar integral injectivity results in Witt theory. This work was sup- ported in part by a grant from the National Science Foundation.
1. The Assembly Map
In constructing the homotopy theoretic analogue of the Baum-Connes assembly map, the ring Sr defined in the introduction plays a crucial role. So we begin with this case.
240 c. OGLE
For 9 s F an element of finite order n, the element
(1 + ~ng + ~Zg2 + ... + ~n-19n-1 ) = ,~=0 ~ingi (1.1)
is an idempotent in the group algebra S r [ F ] , where 4, is an nth root of unity. Denoting this idempotent by e(~,, g), the module
P(~., g) = S r [ r J " e((., 9) (1.2)
is a projective left Sr [F] -module of rank 1. There are n distinct projective modules which arise in this fashion, one for each nth
root of unity. Fixing ~, = e 2"i/", the kth projective module is then p(~k, 9)- For each k, define a representation
p(~k, O): Cg -~ Autsr[vl(P(~k,, 9)),
p( ~ , 9)(h )(a) = a " h. (1.3)
Note that this is a well-defined left S r [F j -module automorphism for one has an equality
r" e(~, k, 9)" h = r" h" ~(~k, 9) (1.4)
if h is in the center of g. Recall that if P is a finitely generated projective module over a ring with unit R, there exists a complimentary projective Q such that P @ Q ~ R N. This determines a map
AutR(P) (I)e(P)* GLN(R)
f ~, ( f E) Ida): R N -~ R N (1.5)
under the identification of R N with P @ Q. This map depends on Q and the choice of isomorphism P 0) Q ~ R N, Changing the isomorphism yields a map conjugate to the one in (1.5). A different choice of Q yields a different map; however because P is finitely generated these maps become conjugate upon stabilization. Therefore, the
stable version of 0.5)
AutR(P) (I)(P) ~ GL(R) (1.6)
is well-defined up to inner automorphism. Let i e ord(g). The composition
A ' ( i , g ) : B C o x {i} B C ° Bp(~,g!) p i BAuts~trl( (~., g)) B¢(P(~"'g)), BGL(Sr [F] ) ) (1.7)
defines a map
A'(g): B C o x ord(g)-+ Ko(Sr[F]) x BGL(Srl-F]) (1.8)
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 241
given by
A'(g)(x, i) -- ( [P (~ , g)], A'(i, g)(x)).
Here [P (~ , g)] denotes the projective class of P(~,, g) in Ko(Sr[F]) . Finally, we pass to K-theory via the plus construction:
A(g): BC o x ord(9) A'(g))Ko(Sr[F]) x BGL(Sr [F ] )
, K o ( S r [ r ] ) x BGL(Sr [F ] ) +
K(Sr[F]). (1.9)
P R O P O S I T I O N 1A0. Up to homotopy and the reorderin 9 of ord(g), A(g) only depends on the conjugacy class of the cyclic subgroup (9).
Proofi If (g') is a cyclic subgroup conjugate to (g), the 3a e F with (g') = (g~)- Conjugation by c~ determines an isomorphism Cg (-Y, Co= and a commuting diagram
Co P(~"'g) , Autsrirl(P(~,g))
i Cg= p(~"' f)~ Autsr[rl(P(~n, g )).
(1.tt)
Co= C o, and ~ ~ = {P(~.,g )}o.<,.<~,~-i = {P(~i,,g )}o.<~<~'E-,-
Mapping the right-hand side of (1.11) to GL(Sr[F] ) , passing to classifying spaces and performing the plus construction produces a commuting diagram (for each i):
{ i } × O C o A'( i '9)+ " B G L ( S r [ F ] ) +
( - ) ' i l (-)" (1.12) {i} x BC o, . A'(i'°~)+ , B G L ( S r [ F ] ) + .
The vertical map on the right is homotopic to the identity. It follows that A(g) and A(g') are homotopic up to a re-ordering of ord(g') corresponding to the choice of g~ vs. g' as the generator at (g'). []
This proposition suggests taking a disjoint union of maps of the form given by (1.9) indexed over the conjugacy classes of cyclic subgroups of F -
A'(F) do~ H A(g): H BCo x ord(g)-~ K(Sr[F]). (1.t3) <(o)) <(o)>
ord(o) < ~ ord(g) < oo
At this point, the definition of A'(F) depends on choosing a representative (g) from each class <(g)>, and a choice of generator for (g), with different choices affecting A'(F) in the manner described above. Thus, at the level of spaces, these choices should be considered as part of the data associated with A'(F). K(Sr[F]) admits a nonconnective delooping K(Sr [F] ) (cf. [Wag]) which is a module spectrum over the
242 c. OGLE
ring spectrum K(Sr) with structure map induced by the usual Loday-Waldhausen product ([Lod])
K(SrEF]) A K(Sr )~ K(SrEF] (~) S t ) ~ K(Sr[F]). (1.14) Z
Adding a disjoint basepoint to the left-hand side of (1.13) which we map to the basepoint (0, .) ~ K(Sr[F]), taking the adjoint and smashing with K(Sr) produces a basepointed map of spectra
/ \
A'(F): ( V V (BCg+)] A K(Sr)-," ~(Sr[F]) . (1.15) \ !gi < 09 ord(g) /
((o))
Denoting the double wedge appearing inside the brackets by X(F), we can rewrite (1.15) as
_A_'(F): ~,(X(F); K(Sr)) --, =:K(Sr[F]). (1.16)
So far we have worked integrally. However, the left-hand side of (1.16) contains a certain amount of redundancy corresponding to the fact the representation p(~,, g) will appear once for every representative (g') containing a conjugate of g- In certain cases, this redundancy can be avoided by making further choices. Over Z these choices are unnatural.
PROPOSITION 1.17. Let g ~ F be offinite order n. Then g can be written as a linear combination
n - 1 t t - 1 g = ~oe(~°,g) + "'" + a,-1~(~, , g )= ~ aie(~i.,g) (I.18)
i = 0
where ~i e Sr. Proof. If M is the matrix with
lvt,j = 1 . ( j_ . ) . n
then M is invertible over St. []
This elementary proposition corresponds to the equally elementary fact that the delta function on the generator t of Z/n occurs as the character of a virtual representation of 7_In over Sz/,. Returning to (1,16), we can alternatively view the spectrum F~(X(F); •(Sr)) as a wedge of spectra indexed over conjugacy classes ((g)). Restricting to a fixed class we get
A'(F)<(o)>: H ( V (BCo)+;~(Sr) / ~ K(Sr[F]). (1.19) \ ord(g) /
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 243
We now pass to homotopy groups and tensor over Z with St, writing H,(g) for H,(BCg,; K(Sr))®z Sr. Consider the map
H,(g) --, H,(g) G... G H(g) = + H,(g)
x ~ (0~oX, 0qx,..., a,,_ ix) (1.20)
where n = ord(g) and al is as in (1.18). Composing with (1.19) we get a homomor- phism of groups
A(F)<g>: H,(g) --* K, (S r [F ] ) @ St. (1.21) g
The argument of Proposition 1.10 implies A(F)<g> = A(F)<g,> if (g) = (g') . Thus, for each elliptic conjugacy class (g), we may choose a representative g and define A(F)<g> in terms of (A'(F)<(g)>), ®z Sr and the homomorphism in (1.20). Summing over elliptic conjugacy classes, we get a well-defined homomorphism
A(F) ace (~ A(F)<o>: (~ H,(BCg;K(Sr)) @ Sr ~ K , (S r [F ] ) (~) St. ( g ) ( g ) z
ord~g) < ~ (1.22)
This is our (algebraic) generalized assembly map for the group F with coefficients in St. The same construction applies for any discrete commutative ring R with unit containing Sr. When necessary, we indicate the dependence on R by the notation
A(F;R): (~ H,(BCo+;K(R))@ Sr+K,(R[FI)@ St. (1.23) <o) z
o rd (g ) < oo
This is one case of interest to which we will return further on. Another case we will consider is that of a complex Banach algebra B(F) which contains the complex group algebra C[F] . The above procedure applies to produce an assembly map
A(F;B): (~ H,(BCo;K(B)) (~) Sr ~ K , ( B ( F ) ) @ St. (1.24) (g) z z
o r d ( g ) < m
Typically, B(F) occurs as a completion of C[F] . The K-theory spectra appearing in (1.24) are the nonconnected Bott-periodic spectra associated with a complex Banach algebra. As a third case, we may consider the complex group algebra C(F), topologized in a way making it a continuous module over C (in order to be interesting, this topology should also allow for the standard embedding C[F] ~ C*F to be continuous). In this case the Bott-periodized spectrum K(C[F])[/~-1] is a module over Kt°p(C) = K(C), and our assembly map is
A(F;C): G H,(BCo;K(C)) @ Sr-+ K,°P(C[r])[fi --1] (~) St. (1.25) (g) ~ z
o r d ( g ) < oo
Suppose that A is a ring with involution. Let KH(A) denote the nonconnective Hermitian K-theory spectrum of A associated to the trivial form e = 1 (cf. [Karl]) .
244 C. OGLE
Recall that the hyperbolic map can be realized as a map of spectra
K(A) ~ KH(A). (1.26)
As a subring of C, Sr is closed under the involution induced by complex conjugation. The adjoint of A'(F) in (1.13) and the hyperbolic map determine a basepointed map of spectra (compare (1.15)):
AH'(F): ( 101< ~o<(o)>V or,iV(0) (BCg+)l/] A KH(Sr)
KH(SrEF]) A KH(Sr) ~ KH(Sr[F]). (1.27)
where the last map is the Loday-Karoubi product on Hermitian K-theory. The above procedure yields an (algebraic) generalized assembly map for Hermitian K-theory:
AH(F) do_~_f (~ AH(F)<0>: (~ H,(BCg; KH(Sr)) (~ Sr <g> <o>
o r d ( g ) < o o
--* KH,(Sr[F]) (~ Sr. Z
(1.28)
One similarly may construct a generalized assembly map for topological Hermitian K-theory. In both cases, the maps commute with the hyperbolic map. Inverting 2, (1.28) therefore induces an assembly map for the localized Witt groups of St [F] :
AW(F) dee (~ AW(F)<g>: (~ H,(BCo;W(Sr)[½]) (~ Sv <g> <o> z
ord (o ) < oo
W,(Sv[F]) ~ 7/[½] (~) St. (1.29) Y Y
By Sullivan localization, these group homomorphisms can be realized on the level of spectra.
At first glance, it may seem that introducing the coefficient ring Sv both inside and out is needed in order to construct a homotopy-theoretic analogue of the BC assembly map. In fact, such a homotopy-theoretic description exists integrally, as we will show in subsequent work.
2. Group Algebras and Cyclic Theory
R in this section will be a commutative ring with unit. All tensor products in this section will be over R. The notation
HH,(A), HC,(A) (2.1)
will denote the Hochschild, resp. cyclic, homology of A (over R). The same notation applies for the corresponding cohomology theories. We consider the Hochschild and
A S S E M B L Y M A P S , K - T H E O R Y , A N D H Y P E R B O L I C G R O U P S 245
cyclic theory of the group ring R[F] . A well-known result is that the Hochschild homology of this ring decomposes as
HH,(R([F]) - I~ H,(BCo; R). (2.2) (g>
Let HC, (R[F] ) be the cyclic homology groups of the (discrete) ring R[F] . Burghelea in [Bul l computed the groups HC,(R[F]) , again in terms of a sum over the conjugacy classes of F. For now we are only interested in the elliptic part, that is, the part corresponding to the conjugacy classes of elements of finite order. Define
H H ~ ( R [ F ] ) = (~ H,(BCo;R ) <O)
ord(g) < co
H C ~ ( R [ F ] ) = (~ H, (BCg;R)@HC, (R) (2.3) <g} R
ord(g) < o~
Then Burghelea's computation implies
THEOREM 2.4 (cf. [Bul]). Assume that the order of each element of finite order in F is invertible in R. Then H C , ( R [ F ] ) ~ HC~(R[F])@ (nonelliptie). Moreover, this splitting is compatible with the Connes-Gysin long-exact sequence:
B B ... , H H , ( R [ F ] ) / - ~ H C , ( R [ F ] ) s HC,_z(R[F]) - - , . . .
0 H H ~ ( R [ F ] ) ~ H C ; ( R [ F ] ) s HC;_2(R[F]) o • *. y.;
(2.4)
In the bottom sequence S can be given in terms of the decomposition of (2.4) as S = Id ® S(R), where S(R):HC,(R) ~ HC,-2(R). []
The same theorem applies to cohomotogy. Let
HH*(R[F]) = I-[ H*(BCg;R), <o>
ord(o) < 0o (2.6)
HC.*(R[F]) = IF[ H*(BC,; R) @ HC*(R). <0> R
ord(o) < co
Then the above theorem admits a dual formulation. Under the same conditions on R, we have
THEOREM 2.7. There is a splitting of the Connes-Gysin sequences:
B* B* HC,_2(R[F] ) s * HC*(REF]) !~, HH*(R[F]) - - , HC*-~(R[F]) ~ . . .
0 S* I* .... --.- HC*-~(R[F]) ~ HC*(R[F]) ,, HH*(R[F]) 0 NC,_t (R[F]) ---,...
(2.8)
246 c, OGLE
where the S* map in the bottom sequence is given by
S* = Id ® S*(R), S*(R): HC*-2(R) ~ HC*(R). []
Formula (2.6) involves products rather than sums as we are working with cochains which may not have compact support (when F is a finitely generated word- hyperbolic group, Rip's theorem implies there are only finitely many elliptic conjugacy classes. Thus, for our applications to hyperbolic groups, the distinction here between sum and product is irrelevant). Theorem 2.7 tells us that for each elliptic conjugacy class (g), there are injections
H*(BCg; R) ~ , HC*(R [r]). -~ / (2.9)
HC*(R[F])
We will need a precise formulation of the horizontal map in (2.9). For this, we begin by fixing a representative g of an elliptic class (O). In what follows, EF will denote the standard simplicial homogeneous bar resolution on F, with F acting on the left. Sg denotes the set of elements in F conjugate to g; the action of F on Sg is given by
x(g') = x - lg'x; x e F, g' e S o.
Finally, for a F-set S, EF x r S is the quotient of EF x S by the relation which in degree n is
([ao,..., a,]; x(s)) = ([xao,..., xa,]; s).
(2.10) The Map N.~r(F)~ EF xrS a. Let gl ® gl ®"" ® g, e NT(F) be an arbitrary n simplex. Map this by
[l, go, gogl .... ,gogl . . .g ,-1]; gi if gi eSa, go ®"" ® g,, ~ \~=o (2.11)
[.0 otherwise.
It is easily seen that this produces a well-defined simplicial map.
(2.t2) The Isomorphism EF xrSg ~ EF xr(Cg\F). An element of Sg is of the form x-~gx, x e F. The map on n-simplices is given by
([O~o, ~ ..... o~,]; x- ~gx) ,-, ([C(o, o~i,..., ~.]; (C,)x).
x~ lgx i = x~lgx2 if and only if g = (xlx-~l)g(xlx~l) -1, implying that x ix~l e Cg, whence (Cg)x2 = (Cg)xl. Thus, the map S o ~ Cg\F given by x - l g x ~ Cgx induces a F-equivariant isomorphism of sets, and so an isomorphism of simplicial sets as indicated by (2.12).
ASSEMBLY
(2.13) The This is the
(E~o,
MAPS, K-THEORY, AND HYPERBOLIC GROUPS
Weak Equivalence EF ×r (Co\F) -~' Ca\EF. simplicial map given on n-simplices by
~ , . . . , ~,,3; (C~)x) = ([X~o, x~ , . . . , x~,]; Cg)
[X~o, x,~,.. . , x~,].
247
(2.14) The Weak Equivalence Cg\EF ~ Cg\ECg. There is a projection F P%; C0\F; fixing a section s of the projection on the level of sets, we consider the simpliciat map E F ~ ECg given by
["o . . . . ,",3 ~ [~,"~,-- - , ~;3,
where ~; = ~i(s(po(,i)))-1. If fl~ = h~, then fl; = h~(S(pg(h~)))-1 = h~;, It follows that the map EF ~ EC o is
equivariant with respect to the left action of Co, so descends to give a simplicial map Co\EF ~ Co\EC o.
To summarize, we have constructed for each g a natural map
N.CY(F)--* Co\EF (2.15)
and an unnatural map
Cg\EF L; Co\ECo (2.16)
depending on a choice of section s for the projection Po, which is a homotopy inverse to the natural weak equivalence Cg\ECa ~ ~- ~ Co\EF going the other way (the choice of section s will be important later on in showing that cyclic group cocycles associated to elliptic conjugacy classes pair with K.(C,(F)) for word-hyperbolic groups F). Note that (2.13) and (2.14) together amount to Shapiro's lemma on the level of spaces,
PROPOSITION 2.17. Let R be a discrete commutative ring with unit containing Sv. Then any eohomology class [c] ~ H"(BCo; R)((g) elliptic) is represented by a cocycIe c: (ECo). = (II" +1Co) -~ R satisfying
(i) C(~o, . . . , ~ , ) = c(h~o, hO~l . . . . , h~,) , h E Co; (ii) c(0¢o .. . . . , , ) = 0 if ~i+1 = gk~ i for some k,O <~ i < n;
(iii) C(~o, ..., ,,,) = 0 / f ~ , = gk~ o for some k.
Proof. (i) is by definition. (ii) and (iii) follow from the dual form of Burghelea's theorem together with the observation that the group epimorphism C o -~ Cg/(g) ((g) is normal in Co) induces an isomorphism H*(B(Ca/(g)); R) - ~ H*(BCa; R). []
View R[F] as the convolution algebra of R-valued functions on F with compact support; fix g and a cocycle representative c ([c] e H"(BCo; R)) normalized in the manner described by (2.17). Define
n + l
~,<o>: ® R[F] ~ R R
248
by
C. O G L E
~,<g>(/o ® i ®--- e L)
= ~ ( ~ OOno, ..... =g°" ..... f,(g~)c(x',(xgo)', .... (xgo ..... g,- 1)')) ).
(2.18)
Here a' and the section s defining it are as in (2.14). The section s' is a potentially distinct section of the projection Po: F ~ Co\F. The need for using distinct sections will become clear in the proof of the cocycle extension theorem given in Section 4.
The composition of maps (2.10)-(2,14) produces a simplicial map N?'(F) Cg\ECg and, hence, a morphism of complexes
) c*(cAEco; R)--, C*(NT(r); R) ~ HomR ® R[r], R R
which sends c E C"(Co\ECo;R) to zc,<o>. Thus, zc,<o> is a Hochschild cocycle; the normalization of c guarantees that ~c,<g> vanishes on im(B)c ®~+~ R[FJ, hence defines a cyclic cocycle.
At the conjugacy class (1), we recover the formula of Jolissaint ([Jol], p. 60; [CM], section 6). For n = 0, ~1,<0> is the trace on the group algebra associated with the conjugacy class (g) :
zl,<o>(fo)= ~] fo(g*). (2.19) x=s'(x') x' e Cg\F
z~,<o > will not have compact support unless C o has finite index in F (this fact is the main obstacle in attempting to show that the elliptic traces extend over Ko(C~F) for (g) # (1)).
We conclude this section by explaining its relation to the previous one. For cyclic homology, the homomorphism
HC~(R[F]) A*nC(F;R)) HC.(R[F]) (2.20)
arising in Burghelea's theorem is the algebraic analogue of the generalized assembly map constructed in Section 1. The cocycle z¢,<o> satisfies two important properties with respect to this map. First, as the support of r~,<o> is concentrated on the set
we see that the composition
H,(BCo.;R)@ HC,(R)),,-->HC~,(R[F1)~HC,(R[F]) [%'<g>]*; R (2.21) R
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 249
is zero if ( g ) # (g ' ) . When ( g ) = (g ' ) , the normalization of the cocycte repre- sentative c guarantees that [zc.(g)] is also zero on the image of the summand (Oi>o Hn-zi(BCg; R)®R HC2~(R)) of HC~(R[F]) under the algebraic assembly map AH, C(F; R). For each integer j >/0, we may define -j re.(0) as the composition
HC.+2j(R[F]) s'J} HC,(R[F]) Dc.<g>], R. (2.22)
Again by the normalization of the cocycle c, we have that the restriction of f{,<o> to the image of (@~>0 H,-2i(BCo; R) ®R HCz(~+j}(R)) under the assembly map is zero. These facts allow an alternative view as to why the algebraic assembly map for cyclic homology is injective (at least when R is a field. This approach will be used later on, in Section 5). Let
0 5 k X = (Xn, X n - 2 . . . . , X n - 2 i , . . . ) ff HC,(R[F])
where
x,-2~6 (~ H,-2 , (BCo;R)@ HC2,(R). (o) R
ord(a) < oo
Choose the smallest integer k 1> 0 such that X,-zk ¢ 0. Then, by the previous arguments and the formula given in (2.18), we have
- k H C . z~,<g>(A,, (F, R)(x,, x~-z .... )) = (c, X,-2k) ~a 0. (2.23)
This argument does not replace Burghelea's, as his result was used in (2.17) to give a quick proof that elliptic group cocycles can be suitably normalized (a fact which is used above in determining the behavior of Q(g>).
3. Hyperbolic Groups and the Harish-Chandra Algebra
We begin with a description of the rapid decay algebra, due to Haagarup [HI (and subsequently studied by Jolissaint and P. DeLaHarpe; cf. [Jol], [Joll]). Beginning with a discrete group F equipped with a word-length function 9 ~ L(g) and a real number s, one defines the Hilbert space H}JF) as
H}JF) = { f : F ~ C I Ilfll2,s,L < oo}. (3.1)
The norm tl f II 2,~,L is the Sobolev norm lJ f ]t 2,~,L = ( ( f , f ) 2,~,L) 1/2, where
( f , f')2,s,L def E f(g)f'(g)(1 + L(g)) 2s. (3,2) geF
Let
H~(F) = 0 H},(F). (3.3) s
Note that each H~(F) can be viewed as the completion of C[F] in the norm [I I]2,s,L; moreover, the countable subcollection of norms {[I 112,..Lfn ~ N} give H~(F) the structure of a Fr6chet space.
250 c. OGLE
The primary obstacle in using this space to study the K-theory of C*F is that it may not be a sub-algebra of C*F (or even an algebra).
DEFINITION 3.4 (Jolissaint). F has property RD (= Rapid Decay) if there exists a word-length function L on r and numbers a, t 1> 0 such that Iifllr ~< a" Ilfllz,t,L for all f e C [ F ] , where [If]It denotes the norm f viewed as an element in c*r .
For such L and F, H~,L(F) lies in C*F for sufficiently large s hence so does
Hf, L(F).
LEMMA 3.5 (see Lemma 5, §1.2 of [Jol]). I/" F has property RD, H~,L(F) is an involutive subalgebra of C* F. []
We now consider a modification of Haagarup's construction. In addition to the above data, we are given a fixed constant C > 0. For f , f ' : F ~ C we define the
pairing ( j , f')2,~,C,L as
( f , f')2,s,C,L = ~ f (g ) f ' (9 ) [ (1 + L(9))(1 + CU°))] 2~. (3.6) g
Then
with
Ilf Ila,,,c.~ d~f ( ( f , f bz,,,c,z)l/2,
H~,c.L(F) = {f: F --* C IItfHz,~,c,L < o0},
H~,c,L(F) = (-] H~,C,L(F). (3.7) s
H~,C.L(F) satisfies many of the same properties as H~,L(F). In particular, it is a Fr6chet space with respect to the norms {tl IIZ,.,C,L}.~n; for C >/1, H~,C,L(F)c H~,r(F) V s, hence also H ~,C,L(F) c H~,L(F). Note that if C > 1, the exponential term C u°) dominates the term inside the sum so that we could have omitted (1 + L(9)) 2s.
F Its inclusion is, however, useful for making the transition from H~,L(F) to Hz,c,L( ) when C 7> 1.
LEMMA 3.8. I f F satisfies property RD, then H ~,C,L(F) is an involutive subal.aebra of C*F for all C >~ 1.
Proof. This follows by the same argument as Jolissaint's lemma 5, §1.2 of [Jol]. For the sake of completeness, we include a proof.
Since F has property RD and H~C,L(F)~-H~,L(F) for C >/1, there exists a constant a > 0 and an s > 0 such that
tl ~ × 0 II ~ <~ a " tt 4~ II 2,~,L " I10 il 2
for all qS, 0 in H~,C,L(F).
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 251
Note that It q5 x !P tl 2 = 11 ~b x ~ !12,o,C,L. Assume C >1 1 is fixed, and that t >_- 0. Then
2 II q~ x 0 II 2,~,c,L
2 = ~ ~ ( h ) O ( h - t g ) (1 + L(#))2'(1 + CL(°)) 2t
~< ~ I4~(h)l(1 + L(h))'(1 + cL(h))qO(h-*0)l(1 + L(h-lg))'(1 + CUh-lg)) ' ;
(3.9)
this again follows by Peetre's inequality, which implies for all g, h e F, t' ~> 0 and C >~ 1 that
(1 + L(gh))C(1 + cL~gh)) t"
~< (1 + L(g))C(1 + C(h))t'(1 + cL(gh)) ''
~< (1 + L(a))*'(1 + L(h))C(1 + cLt~)+L(h)) ~'
~< [(1 + L(a))"(I + CL{g~)*'][(1 + L(h))C(1 + CL(h))*']. (3.10)
Define functions qS,,c(-), Or,c(-) by
qSt,c(h) = ]q~(h)l(1 + L(h))t(1 + cL(h}) ~,
Ot, c(h) = IO(h)l(1 + L(h))'(1 + cL(h)) ', h ~ F. (3.11)
Then
tl ~b,,c 112,=,c,L = I14~ tl2,,+,,c,L,
1t ~',,c II =,=,c,L = II 0 II2,,+,,C,L, (3.12)
whence 49,,c, O,,c e H~,C,L(F). We thus have
1t q~ x ¢ ll~,,,c,~
~< II ~,,c x 0~,c if=, by (3.9)
~< a- II q~,,c tl2,~,L II ~'~.c ih (Remark 2, §1.2 Uol] ) <~ a" If O~,c II 2,~,c,L " !r ~k ,,c 1f2,0,C,L,
<~ a " lf ~ I[ Z,, + t,C,L " t[ 0 112,t,C,L <0% for a l l t>~0.
since C >~ 1
(3.13)
It follows that ~ x ~eH~,c,L(F). Obviously, [f~*i[2,t,c,L= I[~brl2,t,C,L for all choices of t, C, and L. Note also that nothing is lost in assuming t ~> 0, as
H~,C,L(F) = ('1 H~,C,L(F)= ~ H~,C,L(F). [] t ) O m~N
When H~,C,L(F) is contained in C 'F , it is dense in the norm topology as it contains the group algebra C[F] . We will need to know something about the effect
252 c. OGLE
of the inclusion map H~,e,L(F) ~ C*F on topological K-theory. To this end, one could either directly prove that H~,cx(F) is holomorphieally closed in C*F (as Jolissaint does for HT, L(F)), or follow the approach of Connes and Moscovici ([CM], lemma 6.4) by constructing an algebra/Tcx(F) satisfying the following two properties:
(i) ~cx(F) is an involutive subalgebra of C*F which is dense and holomorphically closed, (3.14)
(ii) /~cx(F) c_ H~,c,L(F).
As we shall see, when H~,cx(F)c C*(F), Kt~P(H~,cx(F)) naturally contains K~v(C~F) as a summand. We will use the Connes-Moscovici approach, as it adopts easily to our setting.
LEMMA 3.15. Let F have property RD. Then there exists an algebra /-7/c,L(F ) satisfying (3.14), such that when topologized as a Frdchet subalgebra of H~.cx(F), the inclusion/~c,L(F) ~ ~ C*F is continuous.
Proof. Following the proof of Lemma 6.4 [CM], we begin by defining an unbounded operator Dc on 12(F) given on delta functions by
Dc(~o) = L(g)CL(°)6o, g ~ F.
Then Oc = ad(Dc) is an unbounded operator on the space ~q~(12(F)). For each k >t 0, let
/ tk L(F) = {a ~ C ' F t Ok(a) is bounded}.
Let
&,L(r) = N kL(r). k ~ 0
/~c,L(F) certainly contains the group algebra, as it contains the elements of F. As with the Connes-Moscovici algebra B,/~c,t(F) is a left ideal in ~k~0 Domain ~ , and this intersection is closed under holomorphic functional calculus. This implies /tc,t(F) itself is closed under holomorphic functional calculus. For a ~ C~F we have
(~(a) oI)(ae)= ~a(6e) = Z Ck'Ug~t(g)~a(g)6o, geF
where I denotes the inclusion of C~F in ~(12(F)). Thus a ~/Tc, L(F) implies
CZkUg)L(g)2~la(g)t2 < co Vk >- 0 oeF
which for C ~ 1 implies
(1 + CU~))2k(1 + L(g))Zkta(g)12 = tlall2,~,cx < oo Vk > 0. g~F
Thus /-tcx(F) c H~,c,L(F), as H~c,~.(F) is a Fr6chet algebra with respect to the collection of semi-norms { I1 - II 2,k,C,L}k >~ O. When F satisfies property RD, we already
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 253
know that the inclusion H~,c,L(F)c--~C*F provided by Lemma 3.8 above is continuous, where H~,c,L(F) is given the projective limit topology. This completes the proof. []
Remark 3.16. (i) As in [CM] we could have algebraically tensored C[F3 with N = the algebra
of functions of rapid decay, and worked with the holomorphic closure of C[F] ®.~ ___ C*F Q Jr. This is the natural thing to if one wants holomorphic closure on the level of n x n matrices for all n. Our algebra/fc(F) would then have contained C[F3 ® ~ as a dense subalgebra, with the norm [I- [tz,k,c,L extended to matrices as in ([CM], Lemma 6.4). This certainly can be done; however it is unnecessary for our purposes.
(ii) When H~,L(F) _ C 'F , H~.c,L(F) c_ C*F for all C/> 1. However, the converse may not be true. It seems possible for sufficiently large C that there may be an inclusion H~,c,L(F)__ C*F for groups F having only has nonpositive sectional curvature, as in the case F is a discrete subgroup of a Lie group.
DEFINITION 3.17. F is said to have property MRD (= Modified Rapid Decay) if there exist constants a, t >~ 0, C ~> 1 and a word-length function L such that for alt f e C[F] ,
llfllr ~< a[ffll2,s,c,L (cf. Definition 3.4).
Thus having property MRD implies that there exists C >f 1 with H2~c,L(F) ~ C'F, which is equivalent to requiring H2,s,c,L(F) ~ C*F for sufficiently large s.
As with the algebra H2L(F), having property MRD is closed under various group operations. The following proposition illustrates the point.
PROPOSITION 3.18.
(i) Let F have property MRD, F' a subgroup of F with both F and F' finitely generated. Then F' has property MRD.
(ii) Let F' be a subgroup of F of finite index, with F is finitely generated; then if F' has property MRD, so does F.
Proof. (i) follows by the same argument as in proposition 1 and corollary 2 of [Jol], §2.1. By the closed graph theorem (and the definition of the reduced C* norm), property MRD implies
3C, s/> 0 and word-length function L such that
rt~0 x 0!12 ~< IIq~ll2,~,C,LIl~ll2; ~ , ~ c [ r ] (3.19)
(compare Lemma 3.8 - note that we are not assuming F has property RD). Using this inequality, the proof of proposition 3 [Jol], §2.t goes through verbatim. []
It is equally straightforward to show by the techniques of proposition 5, section 2.2 of [Jol] that
254 c. OGLE
PROPOSITION 3.20. I f F1 and F2 have property MRD, then so does F1 x F2. D
In fact, one can show more generally that if F1, F2 have property MRD, then so does the semi-direct product F1 x~ 172, where e: Fz -~ Aut(F1) has (suitably bounded) exponential growth. This is less restrictive than the requirement that c~ have polynomial growth.
Note that Lemma 3.15 actually applies to groups with property MRD. It therefore follows from a result of Bost [Bo] that for all groups F with property MRD, the inclusion/~c,L(F) ~- > C*F induces an isomorphism in topological K-theory:
* K , (C, F). (3.21)
Remark 3.22. The modification of the Haagarup algebra presented in this section is first-order exponential. We could further modify the algebra to get second-order exponential rapid decay, with C~ "(g) replaced by a term of the form C~ c~)). In fact, one may, by taking the intersection over all exponential orders of the corresponding completions of the group algebra, produce a rapid decay algebra of the generalized exponential type. This algebra satisfies all of the same properties as H~(F) for hyperbolic groups. There is some reason to believe such a construction may be useful for extending the results of this paper to automatic groups.
4. Extending Elliptic Group Cocycles
In this section, we prove the main technical theorem. Currently, it is stated for word-hyperbolic groups, however the result of the following Section 6 will allow us to apply it to a larger class of groups. In the following theorem, l" will denote the complex vector space of m-summable series.
THEOREM 4.1. Let F be a flnitely-generated word-hyperbolic group in the sense of Gromov. Then there exists a constant C >~ 1 such that for each elliptic class (g) and complex cohomology class [(p] e H*(BCo; C) (n >~ 0), [zo,.,] extends to yield a continu- ous cyclic n-eocycIe on the algebra/~c,L(F) with values in the vector space I m, where m = ord(g) (and qo is a normalized representative of [~0]). Said another way, there is an
injection
HC* (C [ r ] ) ~ He*(Hc,r(r)) (4.2)
such that the composition with the natural map
HC*(/TC,L(F)) -~ HC*(C [F]) --- HC*(C [F]) (4.3)
(induced by the inclusion C[F] c-~/TC,L(C[F]) and the projection described in Theorem 2.8) is the identity. The group HC*(C[F]) was defined in Section 2. HC*(Hc,L(F)) denotes the continuous cyclic homology of/~c,L(F) with respect to the Frdchet subalgebra topology.
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 255
Proof. By results of Gersten and Short [GS], we know that the centralizer C(S) of any finite set of elements S in F is again a finitely generated word-hyperbolic group. Thus, by Gromov [Gr], we may assume that [(p] admits a representative q~ whose norm is polynomially bounded by L (in fact, Gromov shows that for n > 1 [~o] is represented by a bounded class. Polynomial growth for the case n = i follows from the case F = Z and finite generation). This means precisely that with respect to a fixed word-length function L on Cg, there exist constants b ~ 1, p ~ N with
l~0([1, hi, h lhz , . . . ,h~h2 . . . . . h~])l ~< b lYI (t + L(h,)) p (4.4) i= l
for all h~,. . . ,h, eF . b and m may vary with L, but they exist for any fixed hyperbolic metric on Cg. We may therefore require that having fixed g, we choose L so that / tc ,L(F) ~ C*F (this will be needed in the next section). Recall that for f o , . . . , f , ~ C [ F ] , the cyclic group cocycle r,p,<g> associated to ~o is given on ()Co ® " ' ® f . ) by (cf. (2.20)):
%,<o>(fo ® f l ® " " ®f . )
- 2 (fo(g'~)f~(g~) . . . . f , (g , ) )cp(x , (xgo) . . . . . (x9ogl .(4.5) x= s'(x') 0~i, ,O,, x ' f fCg\F \ Oi=g
We also recall that the construction of %,<0> depends upon the choice of sections s and s': Co\F ~ F of the projection Po: F --~ Co\F. As we may choose s, we require that it be minimal with respect to the word-length function L already given. Thus, for all x' e Co\F, we have
s(x') = x such that L(x) <% L(y) for all y e F with pg(y) = pg(x). (4.6)
CLAIM 4.7. There exists a constant N1 such that
n - 1
Iq~(x',(xgo)', . . . ,(xgogt . . . . . g0-1)')1 ~<2"b(1 + L(x)) Nt I ] (1 + L(g~')) N~. (4.8) i=0
Proof. Denote the left-hand side of (4.8) by Z. Then (4.4), (4.6) and repeated application of Petre's inequality produce the norm estimates
Z < b(l + L(x'))P(1 + L((xgo)'))" . . . . . (1 + L((xgo . . . . . 0 .-1) ' ))"
~< b(1 + L(x))2~'(1 + L(xgo)) 2p . . . . . (1 + L(xgo . . . . . a n - l ) ) 2p
~< b(1 + L(x))2np(1 + L(go) ) 2nv . . . . . (1 + L ( g n _ l ) ) 2np
~< 2%(1 + L(x))N~(1 + L(g~)) N . . . . . . (t + L(g~_~))N'.
The last line follows from Peetre's inequality applied to the substitution L(g~)<~ 2L(x) + L(g~), where Na = 2(n z + n)p.
256 c. OGLE
CLAIM 4.9. There exists a section s': Cg\F--* F, and constants B, B1 > 0, k e such that for all x' e Cg\F
L(x) % BI(B kL(°~)) where x = s'(x') (4.10)
Proof. According to Gromov (loc. cit.), one can solve the conjugacy problem for a word-hyperbolic group F and, moreover, that the linear isoparametric inequality holds for conjugation in such groups. In terms of word-length estimates, this means that (given L) one can find constants C~, C2 >~ 1 such that if g and 9' are conjugate in F, there exists a sequence of group elements ~a, ~2,..., e, e F satisfying
(4.11) (i) g = al ,g ' = a., and ~ + l = flic~ifl71 with L(fl~) <~ Cl for all 1 ~< i ~< n - 1
(ii) n can be chosen in (i) so that (n - 1) <~ CdL(g) + L(g')).
By (i) we know that there exists fl with g' = flgfl-1, where L(fl) ~< C] -1. Substitut- ing (ii) gives L(fi)<~ C c:(u°)+L(°')). Thus, s' exists with the stated property for B1 = C c~L(~), B = C1 and k = C2.
Let N = mkN~ and g = HT=og~. Substituting L(g ~) <~ Z~=oL(g}') into (4.10) yields
L(x) <~ B] +l f i B kL(qT). i = 0
Fix a total ordering of F (i.e., an isomorphism of sets e: F ~ N ). For each integer k,
let
Oh(F) = {a e FI (a) k}
and set f~ = fi[o~(r) (thus f~ = f l on Oh(F), and is zero elsewhere). With respect to c~, each J )e H~,L(F) may be written as a convergent sequence of group algebra
elements f l = limk f~. In the completed tensor product ®" H~L(F), )Co ® "'" ® f , may be written uniquely (with respect to the lexicographieal ordering on F "+~
induced by ~) as
lim f~o ® . . . ® r k , . d t l , ko,...,kn
with respect to the same ordering, the infinite series %,<~>(Jo .... , f . ) may be written
a s
• .. zc,<o>(fo ,.. . , f k,). (4.12) rc,<o>(fo, , f , ) = lira ko ko,...,kn
Now define functions 4o . . . . . 4, by
¢~(=) = If~(~)l(1 + L(c0)N(1 + BL(')) u,
G(c0 = [f,(~)l(1 + BL(~)) N.
O~<i~<n--1, (4.13)
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC G R O U P S 257
Convergence in I" of the partial sums appearing on the right-hand side of (4.t2) is implied by the following estimates (which are far from being optimal):
lira ko ,* . . , kn
tl ~,<g>(f~o,. k. . - , f . )]t,~
E E x = s ' ( x ' ) go . . . . . O,~ x ' eCo \F I l g i=g
I(fo(g~) . . . . . f , ( g ~ ) ) r t q g ( x ' , ( X g o ) t , . . . , ( X g o g l . . . . . X g n - 1 ) ' ) l m
n - 1
I(fo(g~) . . . . . f.(g.=))l'~(2"b)"( 1 + L(x)) "~N' [ I (1 + L(g~)) "N' i=0
We conclude that %,<g> extends to a continuous cyclic n-cocycle on the algebra H~,L(F) (topologized by the FrSchet topology) with values in l". This completes the proof of Theorem 2.1. []
< ~ . (4,14)
(a and t are as in Definition 3.4)
(2.abB~.+l)N~)m r iim m m J o i, z , ,+~, ' ,~ ,L . . . . . II f . - 1 II 2,,+ N,B.Z IIf. II2,,+N,~,z
(since II#.II~ < [ I f . II2~,~+N,..L)
<- E E x = s'(x') go . . . . . #~, x ' e C g \ F I lg i=g
by (4.7)
< (2"b) m Z Z x = s ' ( x ' ) go . . . . . o j ~
x" eCg\F Ilgi = g
I(fo(gG) . . . . . f . ( g D ) [ " 1 + B"I +1 B kL(°Dj ( I ~ (1 + L(g '~) ) raN' i=0 / \ i = 0
by (4.9)
~< (2 "bB i "+ ' )N ' ) " E Z x = s ' ( x ' ) go . . . . . O,, x 'aCo\F I lg i=g
l(fo(g~) . . . . . f.(g,~))l" (1 + B ug" ))N (1 + L(g~{)) N i= \ i = 0
~< (2"bB(~"+~)N') m ~.. ~ [~o(g~)~a(g~) . . . . . ¢.(g.=)]" x=s ' ( x ' ) go . . . . . O,, x'eCo\F H g i = o
< (2"bB~"+l)N1)"((~o x ~1 x -.- x ~.) x (40 x ~ x . . . x 4.) x .-. x (~o x ~ x -.. x ~.))(1)
m times
< (2"bB(ln+l)N') m fi(~ol x ~ i X , . . X ~.) 2
<% (2.abBr.+ i)N,). II ~o [l~,tm,r" I1 ~ iI%,.,L . . . . . II ~ . - ~ !l~,,,.,Lil ~. i!T
258 c. OGLE
A priori, it would seem that the construction of the extended cocycle depends on the choice of ordering c~. This is not the case, as the argument above shows that the series in (4.12) is not just m-summable but absolutely m-summable (alternative- ly, one could argue that as any two potentially distinct extensions are continuous and agree on a dense subalgebra, they agree everywhere).
When n = 0, we have H°(BCg; C) = C and the only nontrivial possibility for [c] is [c] = 1, up to scalar multiplication. The resulting cyclic o-cocycle zl,<g> on the group algebra is the cyclic trace assciated to the elliptic conjugacy class (g) . Thus, as a particular case of the above theorem, we have
C O R O L L A R Y 4.15. I f F is word-hyperbolic, (O) an elliptic conjugacy class, there exists a constant C and word-length .function L on F such that zl,(o> extends to a cyclic O-cocycle (= trace) on H~,L(F) with values in Im. []
Let HC~P(H~,L(F)) denote the topological cyclic homology of H~,L(F) (i.e., the tensor product has been completed with respect to the countable collection of semi-norms used to define the Fr6chet algebra structure on H~,L(F) - see [CI] , part I, chap. 6; [Karl , sec. 4.1). By theorem 4.1, z~,<g> induces for all integers j i> 0 a homomorphism
t o p co S 0 3 ['CC'<a)] *) I" HC,+2j(Hc,L(F)) , HCt°P(H~L(F)) (4.16)
which we denote by z{,<o>. Let t: C ~ I m denote the standard inclusion. The algebraic assembly map given in (2.20) together with the homomorphism H C . ( C [ F ] ) - o
t o p m H C . (HC.L(F)) induced by the inclusion C[F]~--~H~L(F) yields the generalized assembly map for HC~P(H~,L(F)):
J~c(r; C): HC~(C[r]) --, t op co HC, (Hc,L(F)). (4.17)
We now prove that this assembly map is injective, by the same reasoning given at the conclusion of Section 2. Thus, let
0 # x = (xn, x n - 2 , . . . ) e H C ~ ( C [ F ] ) ,
Xn- 2i ( ~ Hn_21(BCo; C) @ HC2~(C), (4.18) (~; , c
o r d ( g ) < m
and choose k the smallest integer for which Xn-2k =i ~ O. Picking g and [c] e H"-Zk(BCo; C) such that (c, x , -2k) ¢ 0, Theorem 4.1 and the results of Section 2
yield
- k ~ H C . ~,<~>(A~ (r, C)(x,, x~_~,...)) - k H C . = z(~,<~>(A, (r , C)(x, , x , _~ , . . . ) ) )
= t((c, x~-2k)) ¢ 0 in I m (m = ord(g)). (4.19)
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS
5. Proof of Theorem A
259
Assume F is word-hyperbolic. Let ~,2t°Pw'~-, C) be the assembly map given by the composition
@ <g>
ord(g) <
H,(BCa;K(C)) @ C tot, -1 K , (C[F])[fl ] @ C z z
-, K~°P(/~cMF)) ® c ~ K~p(c*r) ® c z z
g'°'(Hg, dF)) ® C. y.
THEOREM 5.1. A~P(F; C) is injective. Proof. We have already shown that the corresponding assembly map for
HC~P(H~,L(F)) is injective, so it will suffice to prove that ~ ° ( F ; C) maps to A~C(F; C) under the Karoubi chern character. In other words, we need to show that
(~ H,(BCo;K~P(C)) @ C ~P(F;C), K~Op(H~L(F)) @ C (g) ~ z
ord(g) < oo ~ l ch [ch
@ H,(BCo; H C ~ P ( C ) ) @ C 3H*C(F;C)~ HCt°V(H~L(F) )
ord(~,) < c~
(5.2)
commutes. This reduces to checking two things - first, that
G H,(BCg; Kb°p(C)) ® C ~;o,(r-c), K~Op(H~,L(r)) ® c (o) z z
ord(g) < ~ ~[ch [ch
@ H,(BC.; HCL°V(C)) @ C "4*~C(F; C), SC,'°,(nc,dr)) ~
ord(g) < oo
(5.3)
commutes; second, that under ch, the Bott generator in K2i(C ) @~ C maps to the generator of HC2~(C). The first point reduces to checking the corresponding diagram commutes with algebraic K-theory replacing topological K-theory, Hoch- schild homology replacing cyclic homology, and the Dennis trace map substituted for ch, and this is clear. The second point has been shown by Karoubi ([Karl, corollary 4.17). []
260 c. OGLE
By (1.29) and naturality of our assembly map, there is a commuting diagram
H,(BCo; W~'g(z)) @ C , w~g(Sr[F]) @ C (g) ~ z
ord(o) < o~ i [
H,(BCo;Wt°p(c)) @ C , W~P(C[F]) @ C (o) z z
oral(g) < ~ i [
H,(BCo; Wt°P(C)) (~) C , W~P(C*F) ~) C (o> z z
• H,(BCo;K~°P(C)) ® C ~,(r:c) ~°p , ,I( , (c~r)® ¢ <a) z
ord(o)<
(5.4)
By Theorem 5.1, the composition going from the upper left to the lower right is injective when F is word-hyperbolic. Therefore,
COROLLARY 5.5. The complexified (generalized) assembly map
{~ H,(BCo;wa'g(7_)) ~) C --~ w~lg(Sr[F]) @ C
ord(g) < c~
is injective for word-hyperbolic groups F. []
Using the computations of Borel [Bor] and techniques discussed in Section 7 below, the result still holds with Walg(Z) replaced on the left by w"~g(Sr).
6. Axioms for Injectivity
Theorem 5.1 is stated and proved under the hypothesis that F be a finitely generated word-hyperbolic group. In this section, we axiomatize the properties required of F in order for the techniques of Theorem 5.1 to apply. These axioms are of first-order exponential type, and certainly not the most general possible (see remark below). As the reader will notice, they are more involved than the (PC)- (RD) criterion of [CM] which implies rational injectivity of At°p(F,C) when
localized at ( I ) .
(6.1) Axioms for F:
(a) (MRD). There exists a constant C and a word-length function L on F such that H~,L(F) is an involutive subalgebra of C*F.
(b) (EEC) (Exponential elliptic cohomology). For each elliptic conjugacy class (g) of F and cohomology class [c] e H*(BCo; C), there exists a representa-
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS 261
tive e of [el whose norm is exponentially bounded by the word-length function L of (a) above (this is more general than (1.4.4) above).
(c) Each C o occurring in (b) is finitely generated. (d) (EBC) (Exponentially Bounded Conjugacies). For each elliptic class (g ) of 17,
there exists constants B, B1 > 0, an integer k > 0 and representative g of (g ) such that for each g '~ F conjugate to g, there exists an x with g ' = g x and L(x) <~ BI(BkL(°~)), for L as in (a).
It is now straightforward to verify
THEOREM 6.2. I f F is a discrete group satisfyin9 (6.1) (a)-(d), then the topological assembly map
At°P(F; C) (~) C: (~) H,(BCg; C) (~) K,(C) ~ K,(C~*F) (~) C
ord(g) < c~
is injective (K.(F) as defined in proof of Theorem 1.5.1). []
Among other things, it is not necessary that the number of elliptic conjugacy classes be finite. It is also not necessary that there exists k, B1, B in (6.1) (d) which work for all (g) ; due to the localized nature of the detection procedure, these are allowed to vary with (g) .
Remark 6.3. Both (6.1) (b) and (6.1) (d) may be generalized to global exponential bounds on L(x) of arbitrarily high order (see Remark 3.22). However, some formula for the growth of L(x) is necessary here before one can do the modification of the Haagarup algebra to get the 'right' algebra for F. H~,L(F) was constructed with (6.1) (d) in mind. By 'right' we mean one that works for all elliptic conjugacy classes (g ) (if one is only interested in the injectivity of At°P(F, C) localized at 1, this whole discussion of bounded conjugacy classes becomes irrelevant).
We wiU say that F satisfies (6.1) if it satisfies the axioms listed there. This class is closed under various operations. In particular,
LEMMA 6.4. / f 1-'i, F2 satisfy (6.t), then so does F1 x Fz. Proof. By (3.20) we know that (a) is satisfied. (b) follows by the Kunneth
theorem; (c) is clear. (d) follows by taking the word-length function on F1 x Fz to be the sum of the word-length functions for F1 and 1"2. D
7. The Algebraic Case
In this section, R denotes the ring of integers of a number field. Our object is to prove Theorem C. In fact, the results of this section apply to all discrete groups F satisfying the axioms listed in the previous section - an assumption we make of F throughout this section.
Let rl, resp. r2, denote the number of real, resp. complex, places of R. By the results of Borel [Bor], completion at these distinct places induces a ring
262 c. OGLE
homomorphism
which is injective in rational algebraic K-theory; moreover the image of K~,~g(R) ®z Q in K~g(c *' +~2) ®z Q lies in the additive subspace spanned by the Borel generators
~ i , 2 j + i ~ 2 j + l t ~ ~ l<~i<~rx+r2" 1 ~ j < oo
As K~)g(c r' +':) "" • r' +r2 K.Ig(c) Section 1 reduces the proof of Theorem C to showing that the composition
(~ H,(BCg; E,) (~) C -+ (~) H,(BC~; K:g(C) ? Q) @ C {g} Q {g) Q
ord(g) < ~ oral(g) < oo
-.-> K~,'g(C[F]) (~) C z
naturality of the assembly map constructed in
(7.1)
is injective, where E, is the additive subspace of K~lg(c) ®z Q spanned by the Borel generators { Z zp+ I } 1 <~ p < ~ .
The standard involution E2igi ~EZ/gi -1 on C[F] extends to H~,L(F). These involutions induce involutions on K-theory. We adopt the notation K,(A) +, resp. K,(A)- , (HC,(A) +, resp. HC,(A)-) for the positive, resp. negative, eigenspace of the rationalized K groups (cyclic homology groups) of A under the involution induced by the specified involution on A. If G is a discrete group, we define an involution on H,(BG; Q) by
x ~ ( - - 1)"x i fx~H,(BG;Q),n>~l.
We also equip E, with the involution induced on generators by
tzp+l ~- - t2v+l , 1 ~<p< ~ .
The involutions induce one on the groups appearing on the left-hand side of (7.1), given as the tensor product of the two, before tensoring with C. After tensoring with C (and taking the standard involution on this factor), the composition in (7.1) is involution-preserving. However, in order to apply Karoubi's results to the situation at hand, we need to work (at least initially) over a totally real subfield of C. We recall briefly the techniques used by Karoubi in [Kar] to extend the Borel regulator maps for K~lg(C). Given a complex C-algebra with involution (for example, a dense, holomorphically closed and involution-closed subalgebra of a C* algebra), the topological K-groups K~°P(A) are invariant under the involution induced by that on A. That is,
K P(A) -- K ;P(At ÷.
ASSEMBLY MAPS, K-THEORY, A N D H Y P E R B O L I C G R O U P S 263
Suppose, additionally, that A is a Fr6chet algebra. A fundamental result due to Karoubi is that there is a commuting diagram (compare [Karl 6.24)
. . . . . . . , K~p+x(A) ~.+1 , K~e,(A) , K~g(A)_ ; K~V(A) ~ ...
. . . . ,HC,+I(A) s,HC._I(A)3- HH.(A) Z,HC,(A)-----,..-
Here ch, = c h ~ p is the topological chern character of Karoubi dual to the pairing map of Connes ([C1], chap. 1), ch~ ~l the relative chern character ([Karl, chap. 6; see also [CF]), and D, the Dennis trace. The sequence on the top is the obvious long-exact sequence in homotopy induced (at least in positive dimensions) by the homotopy-fibration sequence of spectra
K r e l ( A ) --~ K a l g ( A ) --~ KtOp(A),
while the bottom sequence is the topological Connes-Gysin sequence. Moreover ([Kar], (6.32)), the image of the composition
Kt°p, + lt-J~A' _~._5!.~ K~¢,(A) chT'. HC, - I(A) (7.3)
lies in the positive (resp. negative) eigenspace of HC,_ I(A) (under the involution induced by A) for n odd (resp, even); this yields the Borel-Karoubi regulator
alg - ~ rel - c h ~ + 1 K2,+I(A) (= K2,+I(A) , HC2,(A)-, (7.4)
defined for all n >~ 0. According to ([KarJ, 6.31), when A = C the induced map
is injective on the infinite cyclic subgroup generated by the Boret generator ~2,-~ 6 K~, g- t(C). Now fix g, with p = ord(g). We recall from Section 1 that each representa-
U tion p( v, g) induces a homomorphism
A'(i, g),: H,(BCo; Q) ~ K~g(C[F]) @ Q. (7.5) z
This image lies in K~g(C[F]) + (resp. K,tg(C[F]) -) for • even (resp. odd), for all 0.N<i~<(p-1). Therefore, taking the product with the Borel regulator ~2,,+1 produces a homomorphism
H2n(BC,; Q) --~ H2.(BC,; (E2m+ 1)) _+ K~l(gn + m) + l ( c [ r ] ) - a lg oo - K2(,+,,)+ I(Hcx(F)) . (7.6)
Tensoring with C we get
Hz.(BCo; Q) @ C ~-~ H2.(BCo; (Ez.,+~)) @ C O
- K2(.+.o+I(Hc,L(F)) @ C. (7.7) q Q
264 c. OGLE
which amounts to a factorization of this part of the complexified assembly map of Section 1 through K~lg(C[F]) - ®eC. Therefore, taking the appropriate linear combination as in (1.20), (1.21) produces the factorization of A(F)<g> by the same sequence appearing in (7.7). Following this by the compexification of the regulator map yields the composition
H2.(BC,; Q) (~) C ~ H2n(BCo;(E2m+I)) @ C Q 0
o O - - HC2(.+,n)(Hc ~.(F)) (~) C --* HCE(n+m)(H~L(F)) (~ C
---> oo HC2(,+MHc,dF)). (7.8)
By (6.28), (6.32) of [Kar] and the results of the first two sections, we see that this map agrees (up to multiplication by a nonzero real number) with the composition
H2n(BCo; C) ~-~ HC~(, + m)(C [F]) /~2B(C+rnI(F; C)~ HC2(, + m)(H~,L(F)). (7.9)
~HC , Now A. (F, C) was shown to be injective at the end of Section 4. We may conclude
that for all n, rn ~> 0,
(H,(BCo;E,))2(.+m)+I (~ C --~ galg 2(,+,,)+ I(C[F]) @ C (7.10) <0> ~ z
ord(o) < ce
is injective, which accounts for half of the homology occurring as the left-hand side of (7.t). To show that the map is injective on even-dimensional homology, we use a standard shifting argument, replacing F with F x 77 (this has the effect of replacing each C o with C o x Z). By Lemma 6.4 and the above, we see that the bottom horizontal map in the commutative diagram
(~ (H.(BCo; E.))2(n+m ) @ C . alg " Kz(n+m)(C[F]) @ C <o> z
ord(g) < e~ I [
alg ((" F F (H,(B(Cgx7/);E,))2(.+m)+I(~C , .~ 2(.+,.)+ 1,vL x 2£])@ C (g) z z
o r d ( g ) < co
(7.11)
is injective, implying that the top horizontal map is as well. This completes the proof
of Theorem C.
References
[BC] Baum, P. and Connes, A.: Chern character for discrete groups, in A F&e of Topology, Academic Press, New York, pp. 162-232.
[BDO] Baum, P., Davis, M. and Ogle, C.: The Novikov conjecture for proper actions of discrete groups (in preparation).
[BHO] Baum, P., Higson, N. and Ogle, C.: Equivalence of assembly maps (in preparation).
ASSEMBLY MAPS, K-THEORY, AND HYPERBOLIC GROUPS
[Bor]
[BuU [CM]
[c i ] [CK]
[F J] [GS] [Gr]
[Jol]
[Joll]
[Karl [Ka]
[Lod]
[QI] [Q2]
[Wag]
265
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