THE K-THEORETIC FARRELL-JONES CONJECTURE FOR
HYPERBOLIC GROUPS
ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
Abstract.We prove the K-theoretic Farrell-Jones Conjecture for hyperbolic
groups with (twisted) coefficients in any associative ring with unit.
Introduction
The main result of this paper is the following theorem.
Main Theorem. Let G be a hyperbolic group. Then G satisfies the K-theoretic
Farrell-Jones Conjecture with coefficients, i.e., if A is an additive category *
*with
right G-action, then for every n 2 Z the assembly map
(0.1) HGn(EVCycG; KA ) ! HGn(pt; KA ) ~=Kn(A *G pt)
is an isomorphism. This implies in particular that G satisfies the ordinary Far*
*rell-
Jones Conjecture with coefficients in an arbitrary coefficient ring R.
Some explanations are in order.
Basic notations and conventions. Hyperbolic group is to be understood in the
sense of Gromov (see for instance [12], [14], [33], [34]).
K-theory is always non-connective K-theory, i.e., Kn(B) = ssn(K-1 B) for an
additive category B and the associated non-connective K-theory spectrum as con-
structed for instance in [49].
We denote by VC ycthe family of virtually cyclic subgroups of G. A family F
of subgroups of G is a non-empty collection of subgroups closed under conjugati*
*on
and taking subgroups. We denote by EF G the associated classifying space of the
family F (see for instance [45]).
A ring is always understood to be a (not necessarily commutative) associative
ring with unit.
The K-theoretic Farrell-Jones Conjecture with coefficients. Given an ad-
ditive category A with right G-action, a covariant functor
KA :OrG ! Spectra, T 7! K-1 (A *G T )
is defined in [7, Section 2], where OrG is the orbit category of G and Spectrai*
*s the
category of spectra with (strict) maps of spectra as morphisms. To any such fun*
*ctor
one can associate a G-homology theory HGn(-; KA ) (see [19, Section 4 and 7]). *
*The
assembly map for a family F and an additive category A with right G-action
(0.2) HGn(EF G; KA ) ! HGn(pt; KA ) ~=Kn(A *G pt)
is induced by the projection EF G ! pt onto the space pt consisting of one
point. The right hand side of the assembly map HGn(pt; KA ) can be identified
____________
Date: January 16, 2007.
2000 Mathematics Subject Classification. 19Dxx, 19A31,19B28.
Key words and phrases. Algebraic K-theory of group rings with arbitrary coef*
*ficients, Farrell-
Jones Conjecture, hyperbolic groups.
1
2 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
with Kn(A *G pt), the K-theory of a certain additive category A *G pt. We say
that the K-theoretic Farrell-Jones Conjecture with coefficients for a group G h*
*olds
if the map (0.2)is bijective for F = VCyc, every n 2 Z and every additive categ*
*ory
A with right G-action.
The original K-theoretic Farrell-Jones Conjecture. If A is the category of
finitely generated free R-modules and is equipped with the trivial G-action, th*
*en
ssn(KA (G=G)) ~=Kn(RG) and the assembly map becomes
(0.3) HGn(EVCycG; KR ) ! HGn(pt; KR ) ~=Kn(RG).
This map can be identified with the one that appears in the original formulation
of the Farrell-Jones Conjecture [28, 1.6 on page 257], compare [37]. So the Main
Theorem implies that the K-theoretic version of the Farrell-Jones Conjecture is
true for hyperbolic groups and any coefficient ring R.
The benefit of the K-theoretic Farrell-Jones Conjecture is that it computes
Kn(RG) by a G-homology group which is given in terms of Kn(RV ) for all V 2
VC yc. So it reduces the computation of Kn(RG) to the one of Kn(RV ) for all
V 2 VC yc together with all functoriality properties coming from inclusion and
conjugation.
Let ff: G ! aut(R) be a homomorphism with the group of ring automorphisms
of R as target. Let RffG be the associated twisted group ring. Then one can def*
*ine
an additive category A(R, ff) such that Kn(A(R, ff)*G G=H) ~=Kn(Rff|HH), see [7,
Example 2.6]. The assembly map in the K-theoretic Farrell-Jones Conjecture with
coefficients in A(R, ff) has as target Kn(RffG).
Farrell-Jones [28] formulate a fibered version of their conjecture which has*
* much
better inheritance properties. It turns out that the version of the Farrell-Jo*
*nes
Conjecture with coefficients as formulated in the Main Theorem is stronger than
the fibered version and has even better inheritance properties (see [7, Section*
* 4]).
The case of a torsionfree hyperbolic group. Suppose that G is a subgroup
of a torsionfree hyperbolic group and R is a ring. Then the Main Theorem implies
for all n 2 Z the existence of an isomorphism, natural in R,
M ~=
Hn(BG; KR) (NKn(R) NKn(R)) -! Kn(RG),
(C)
where Hn(BG; KR ) is the homology theory associated to the (non-connective)
K-theory spectrum KR of R evaluated at the classifying space BG of G, (C)
runs through the conjugacy classes of maximal infinite cyclic subgroups of G and
NKn(R) denotes the nth Bass-Nil-group of R. This follows from [9, Theorem 1.3]
and [45, Theorem 8.11]. If R is regular, then NKn(R) = 0 for n 2 Z and
Kn(R) = ssn(KR) = 0 for n -1.
Previous results. A lot of work about the Farrell-Jones Conjecture has been done
during the last decade. Its original formulation is due to Farrell-Jones [28, 1*
*.6 on
page 257]. Celebrated results of Farrell and Jones prove the pseudo-isotopy ver*
*sion
of their conjecture for certain classes of groups, e.g., for any subgroup G of *
*a group
such that is a cocompact discrete subgroup of a Lie group with finitely many
path components (see [28, Theorem 2.1]). The pseudo-isotopy version implies the
K-theoretic Farrell-Jones Conjecture for R = Z and n 1 and the rational K-
theoretic version for R = Z and all n 2 Z. For more explanations, information
about the status and references concerning the Farrell-Jones Conjecture we refe*
*r to
the survey article [46].
Most of the results about the K-theoretic version of the Farrell-Jones Conje*
*cture
deal with dimensions n 1 and R = Z. The first result dealing with arbitrary
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 3
coefficient rings R appear in Bartels-Farrell-Jones-Reich [3], where the K-theo*
*retic
Farrell-Jones Conjecture was proven in dimension 1 for G the fundamental group
of a negatively curved closed Riemannian manifold. In Bartels-Reich [8] this re*
*sult
was extended to all n 2 Z. In this paper we replace the condition that G is the
fundamental group of a negatively curved closed Riemannian manifold by the much
weaker condition that G is hyperbolic in the sense of Gromov, and also allow tw*
*isted
coefficients.
Further results. We mention that the Main Theorem implies that the K-theoretic
Farrell-Jones Conjecture with coefficients in any ring R holds not only for hyp*
*erbolic
groups but for instance for any group which occurs as a subgroup of a finite pr*
*oduct
of hyperbolic groups and for any directed colimit of hyperbolic groups (with not
necessarily injective structure maps). Such groups can be very wild and can have
exotic properties (see Bridson [13] and Gromov [36]). This follows from some ge*
*neral
inheritance properties. All this will be explained in Bartels-L"uck-Reich [6] *
*and
Bartels-Echterhoff-L"uck [4], where further classes of groups are discussed, fo*
*r which
certain versions or special cases of the K-theoretic Farrell-Jones Conjecture h*
*old.
Applications. In order to illustrate the potential of the K-theoretic Farrell-J*
*ones
Conjecture we mention some conclusions. We will not try to state the most gener*
*al
versions. For explanations, proofs and further applications in a more general c*
*ontext
we refer to [6].
In the sequel we suppose that G satisfies the K-theoretic Farrell-Jones Conj*
*ecture
for any ring R, i.e., the assembly map (0.3)is bijective for every n 2 Z and ev*
*ery
ring R. Examples for G are subgroups of finite products of hyperbolic groups. T*
*hen
the following conclusions hold:
o Induction from finite subgroups for the projective class group.
If R is a regular ring and the order of any finite subgroup of G is inv*
*ertible
in R, then the canonical map
colimH G,|H|<1K0(RH) ! K0(RG)
is bijective.
If R is a skew-field of prime characteristic p, then the canonical map
colimH G,|H|<1K0(RH)[1=p] ! K0(RG)[1=p]
is bijective.
o Bass Conjectures.
The Bass Conjecture for commutative integral domains holds for G, i.e.,
for a commutative integral domain R and a finitely generated projective
RG-module P its Hattori-Stallings rank HS (P )(g) evaluated at g 2 G is
trivial if g has infinite order or the order of g is finite and not inv*
*ertible in
R.
The Bass Conjecture for fields of characteristic zero holds for G, i.*
*e.,
for any field F of characteristic zero the Hattori-Stallings rank induc*
*es an
isomorphism
~=
K0(F G) Z F -! classF(G)f
to the F -vector space of functions G ! F which vanish on elements of
infinite order, are constant on F -conjugacy classes and are non-trivia*
*l only
for finitely many F -conjugacy classes.
o Bass-Nil-groups and homotopy K-theory.
If R is a regular ring and the order of any finite subgroup of G is inv*
*ertible
4 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
in R, then the Bass-Nil-groups NKn(RG) are trivial and the canonical
map ~
Kn(RG) =-!KHn(RG)
to the homotopy K-theory of RG in the sense of Weibel [58] is bijective
for every n 2 Z.
o Kaplansky Conjecture for prime characteristic.
Suppose that R is a field of prime characteristic p or suppose that R is
a skew-field of prime characteristic p and G is sofic. (For the notion *
*of a
sofic group we refer for instance to [21]. Every residually amenable gr*
*oup
is sofic.) Moreover, assume that every finite subgroup of G is a p-grou*
*p.
Then RG satisfies the Kaplansky Conjecture, i.e., 0 and 1 are the only
idempotents in RG.
Now suppose additionally that G is torsionfree. Then:
o Negative K-groups.
Kn(RG) = 0 for any regular ring R and n -1.
o Projective class group.
The change of rings map K0(R) ! K0(RG) is bijective for a regular ring
R. In particular Ke0(ZG) = 0. Hence any finitely dominated connected
CW -complex with G as fundamental group is homotopy equivalent to a
finite CW -complex.
o Whitehead group.
The Whitehead group Wh (G) is trivial. Hence any compact h-cobordism
of dimension 6 with G as fundamental group is trivial.
o Kaplansky Conjecture for characteristic zero.
If R is a field of characteristic zero or if R is a skew-field of chara*
*cteristic
zero and G is sofic, then RG satisfies the Kaplansky Conjecture.
Searching for counterexamples. There is no group known for which the Farrell-
Jones Conjecture, the Farrell-Jones with coefficients or the Baum-Connes Conjec-
ture is false. However, Higson, Lafforgue and Skandalis [39, Section 7] constr*
*uct
counterexamples to the Baum-Connes-Conjecture with coefficients, actually with a
commutative C*-algebra as coefficients. They describe precisely what properties*
* a
group must have so that it does not satisfy the Baum-Connes Conjecture with
coefficients. Gromov [36] constructs such a group as a colimit over a direct*
*ed
system of groups {Gi | i 2 I} for which each Gi is hyperbolic. It will be shown
in [4] that the Main Theorem implies that the Farrell-Jones Conjecture with coe*
*ffi-
cients in any ring holds for . It will also be shown that the Bost Conjecture *
*with
coefficients in a C*-algebra holds for .
Controlled topology. A prototype of a result involving controlled topology and
showing its potential is the ff-Approximation Theorem of Chapman-Ferry (see [18*
*],
[31]). It says, roughly speaking, that a homotopy equivalence f :M ! N between
closed manifolds is homotopic to a homeomorphism if it is controlled enough over
N, i.e., there is a homotopy inverse g :N ! M such that the compositions f O g
and g O f are close to the identity and homotopic to the identity via homotopies
whose tracks are small. Here "close" and "small" are understood to be measured *
*in
N considered as a metric space. In particular it says that a homotopy equivalen*
*ce
which is controlled enough represents the trivial element in the Whitehead grou*
*p.
Controlled topology and its variations have been important for a number of
further celebrated results in geometric topology. Some of these are concerned w*
*ith
the Novikov conjecture [17], [22], [32], [41], [59], ends of maps [52], [53], c*
*ontrolled
h-cobordisms [2], [54], Whitehead groups and lower K-theory, [23], [24], [25], *
*[30],
[42], topological rigidity [26], [27], [29], [30], homology manifolds [15], par*
*ametrized
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 5
Euler characteristics and higher torsion [20] and topological similarity [38]. *
* Of
course this list is not complete.
A key theme in controlled topology is to associate a size to geometric objec*
*ts
and then prove that objects of small size are trivial in an appropriate sense. *
*Such
a result is sometimes called a stability or squeezing result. A good example is*
* the
ff-approximation theorem mentioned above. Related is the reformulation of the
assembly maps into a "forget control" version, i.e., the domain of the assembly
map is described by objects whose size is very small while the target is descri*
*bed
by bounded objects. This formulation of forget-control is often referred to as *
*the
"-version. Now it is clear what one has to do to prove for instance surjectivit*
*y, one
must be able to manipulate a representative of an element in K-theory so that it
becomes better and better controlled without changing its K-theory class. This
opens the door to apply geometric methods. In their celebrated work Farrell-Jon*
*es
used three decisive ideas to carry out such manipulations: transfers, geodesic *
*flows
and foliated control theory.
There is also a somewhat different approach to the assembly map as a forget-
control map, sometimes called the bounded or categorical version. Here the em-
phasis is not on single objects and their sizes but on (the category of) all bo*
*unded
objects. Then the way boundedness is measured can be varied, for instance on
non-compact spaces very different metrics can be considered. A good example is
the description of the homology theory associated to the K-theory spectrum of a
ring in [50]. This formulation is very elegant, but less concrete (and involves*
* usually
a dimension shift).
Controlled topology is the main ingredient in proofs of the Farrell-Jones Co*
*n-
jecture, whereas for the Baum-Connes Conjecture the main strategy is the Dirac-
Dual-Dirac-method.
A rough outline of the proof. We will use the bounded (more precisely, the con-
tinuous controlled) version of the forget-control assembly map. This quickly le*
*ads
to a description of the homotopy fiber of the assembly map as the K-theory of a
certain additive category, see Proposition 3.8. We call this category the obstr*
*uc-
tion category. A somewhat artificial construction makes the obstruction category
a functor of metric spaces with G-action, see Subsection 3.4. In the simplest c*
*ase
the metric space in question is the group G equipped with a word metric, but it
will be important to vary the metric space._This_will be done in two steps. Fir*
*stly,
we use a transfer to replace G by Gx X , where X is a compactification_of the R*
*ips
complex for G, see Theorem 6.1. The benefit of the G-space X is to have place f*
*or
certain equivariant constructions which cannot be carried out in G itself. In p*
*ar-
ticular, in [5] we constructed certain G-invariant open covers, see Assumption *
*1.4.
The existence of these covers can be viewed as an equivariant version of the fa*
*ct
that hyperbolic groups have finite asymptotic dimension._Secondly, we apply con-
tracting maps associated to open covers of Gx X , see Proposition 5.3. This map
will_only be contracting with respect to the G-coordinate and will expand in the
X coordinate. This defect can be compensated,_because the transfer produces ar-
bitrary small control with respect to the X -coordinate. Improving on an idea f*
*rom
[10] we formulate and prove a kind of stability result for the obstruction cate*
*gory
in Theorem 7.2. This result is not formulated in terms of single elements, but *
*as
a K-theory equivalence of certain categories. (However, for K1 it is not hard *
*to
extract a more concrete statement along the lines of the above stability statem*
*ents,
see [10, Corollary 4.6].) The general strategy of the proof is worked out in Se*
*ction 4,
see in particular Diagram (4.4).
Our approach is very much influenced by the general strategy of Farrell-Jone*
*s.
However, our more general setting involves new ideas and techniques. We prove
6 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
the K-theoretic Farrell-Jones Conjecture for arbitrary coefficient rings and al*
*so
for higher K-theory. We also would like to mention that our proof unlike many
other proofs treats the surjectivity and injectivity part simultaneously. One m*
*ain
difficulty is that we cannot work with manifolds and simplicial complexes anymo*
*re
and do not have transversality or general position arguments at hand, since in *
*the
world of hyperbolic groups we can at best get metric spaces with very complicat*
*ed
compactifications. This forces us to use open covers. A benefit of our approa*
*ch
is that we avoid the hard foliated control theory. Other ingredients of the Far*
*rell-
Jones strategy are still used. Namely, in order to show that hyperbolic groups *
*fulfill
Assumption 1.4 we build in [5] on Mineyevs [48] replacement of the geodesic flow
and generalize the long and thin cells of Farrell-Jones for manifolds to certai*
*n covers
of metric spaces.
Open problems. There is an L-theoretic version of the Farrell-Jones Conjecture.
An obvious problem is to extend our methods for K-theory to L-theory. The main
difficulties concern the transfer and the fact that in L-theory one needs to co*
*ntrol
the signature of the fiber and not - as in K-theory - the Euler characteristic.
If both the K-theoretic and the L-theoretic Farrell-Jones Conjecture hold for
R = Z as coefficients for a group G, then the Borel Conjecture is true for G, i*
*.e.,
if M and N are closed aspherical topological manifolds of dimension 5 whose
fundamental groups are isomorphic to G, then M and N are homeomorphic and
every homotopy equivalence M ! N is homotopic to a homeomorphism.
Another problem is to prove the Farrell-Jones Conjecture with coefficients f*
*or
groups which act proper and cocompactly on a CAT(0)-space.
Acknowledgements. The authors are indebted to Tom Farrell for fruitful dis-
cussions and sharing his ideas. The work was financially supported by the Son-
derforschungsbereich 478 - Geometrische Strukturen in der Mathematik - and the
Max-Planck-Forschungspreis of the second author.
1.Axiomatic formulation
Theorem 1.1 (Axiomatic Formulation). Let G be a finitely generated group. Let
F be a family of subgroups of G. Let A be an additive category with right G-act*
*ion.
Suppose
(i)There exists a G-space X such that the underlying space X is the realiz*
*ation
of an abstract simplicial_complex;
(ii)There exists a G-space X which_contains X as an open G-subspace such
that the underlying space of X is compact, metrizable and contractible;
(iii)Assumption 1.2 holds;
(iv)Assumption 1.4 holds for F.
Then for every m 2 Z the assembly map
HGm(EF G; KA ) ! Km (A *G pt)
is an isomorphism.
Sections 3 to 7 are devoted to the proof of Theorem 1.1. The general structu*
*re of
the argument is described in Subsection 4.4. We now formulate the two assumptio*
*ns
that appear in Theorem 1.1.
__
Assumption_1.2 (Weak Z-set condition)._There exists a homotopy H :X x[0, 1] !
X , such that H0 = id_Xand Ht(X ) X for every t > 0.
In order to state the second assumption we introduce the notion of an open
F-cover.
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 7
Definition 1.3. Let Y be a G-space. Let F be a family of subgroups of G. An
open F-cover of Y is a collection U of open subsets of Y such that the following
conditions are satisfied:
S
(i)Y = U2U U;
(ii)For g 2 G, U 2 U the set g(U) := {gx | x 2 U} belongs to U;
(iii)For g 2 G and U 2 U we have g(U) = U or U \ g(U) = ;;
(iv)For every U 2 U, the subgroup {g 2 G | g(U) = U} lies in F.
Suppose U is an open F-cover. Then |U|, the realization of the nerve, is a
simplicial complex with cell preserving simplicial G-action and hence a G-CW
complex. (A G-action on a simplicial complex is called cell preserving if for e*
*very
simplex oe and element g 2 G such that the intersection of the interior oeO of *
*oe with
goeO is non-empty we have gx = x for every x 2 oe. Notice that a simplicial act*
*ion is
not necessarily cell preserving, but the induced simplicial action on the baryc*
*entric
subdivision is cell preserving.) Moreover all its isotropy groups lie in F. R*
*ecall
that by definition the dimension dim U of an open cover is the dimension of the
CW -complex |U|.
If G is a finitely generated discrete group, then dG denotes the word metric*
* with
respect to some chosen finite set of generators. Recall that dG depends on the
choice of the set of generators but its quasi-isometry class is independent of *
*it.
Assumption 1.4 (Wide_open F-covers). There exists N 2 N, which only depends
on the_G-space X , such that for every fi 1 there exists an open F-cover U(fi*
*) of
Gx X with the following two properties:
__
(i)For every g 2 G and x 2 X there exists U 2 U(fi) such that
{g}fix{x} U.
Here {g}fidenotes the open fi-ball around g in G with respect to the wo*
*rd
metric dG , i.e., the set {h 2 G | dG (g, h) < fi)};
(ii)The dimension of the open cover U(fi) is smaller than or equal to N.
We remark that if Assumption 1.4 holds, then it is possible to massage the c*
*overs
U(fi) (using for example Lemma 5.1) in order to additionally_obtain the property
that each U(fi) is locally finite, i.e., every point in Gx X has a neighborhood*
* U that
intersects only a finite number of members of U. We will however not use this f*
*act.
2. The case of a hyperbolic group
Lemma 2.1. Let G be a word-hyperbolic group. Then the assumptions appearing
in Theorem 1.1 are satisfied for the family F = VCyc of virtually cyclic subgro*
*ups
of G.
Proof.(i)Fix a set of generators S. Equip G with the corresponding word metric.
Choose ffi such that G becomes a ffi-hyperbolic space. Choose an integer d > 4f*
*fi + 6.
Let Pd(G) be the associated Rips complex. It is a finite-dimensional contractib*
*le
locally finite simplicial complex. The obvious simplicial G-action on Pd(G) is *
*proper
and cocompact. In particular Pd(G) is uniformly locally finite and connected. I*
*ts
1-skeleton is the Cayley graph of G with respect to the set of generators consi*
*sting
of non-trivial elements in the ball of radius d about the identity in G. All t*
*hese
claims are proven for instance in [14, page 468ff]. Since the quasi-isometry t*
*ype
of the Cayley graph of a group is independent of the choice of the finite set of
generators, the 1-skeleton of Pd(G) with the word metric is a hyperbolic metric
space. Hence Pd(G) equipped with the word metric is a hyperbolic complex in the
sense of Mineyev [48, page 438]. We take X = Pd(G).
8 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
We mention that Pd(G) is quasi-isometric to the Cayley graph of the group.
Moreover, the barycentric subdivision of Pd(G) is a G-CW -complex which is for
large enough d a model for the classifying space for proper G-actions (see [47]*
*), but
we will not use_this fact.
(ii)We take X = X [ @X to be the compactification of X in the sense of Gromov
(see [34], [14, III.H.3]).
__
(iii)According to [11, Theorem 1.2] the subspace @X X satisfies the Z-set con-
dition. This implies our (weaker) Assumption 1.2 which is a consequence of part
(2) of the characterization of Z-sets before Theorem 1.2 in [11].
(iv) This assumption is proved in [5, Theorem 1.2].
Because of Lemma 2.1 the Main Theorem follows from Theorem 1.1. The re-
mainder of this paper is devoted to the proof of Theorem 1.1.
3. Controlled algebra and the fiber of the assembly map
3.1. A quick review of controlled algebra. Let Y be a space and let A be a
small additive category. Define the additive category
C(Y ; A)
as follows. An object is a collection A = (Ax)x2Y of objects in A which is loca*
*lly
finite, i.e., its support supp(A) := {x 2 Y | Ax 6= 0} is a locally finite subs*
*et of
Y . Recall that a subset S Y is called locally finite if each point in Y h*
*as an
open neighborhood U whose intersection with S is a finite set. A morphism OE =
(OEx,y)x,y2Y:A = (Ay)y2Y ! B = (Bx)x2Y consists of a collection of morphisms
OEx,y:Ay ! Bx in A for x, y 2 Y such that the set {x | OEx,y6= 0} is finite for*
* every
y 2 Y and the set {y | OEx,y6= 0} is finite for every x 2 Y . Composition is gi*
*ven by
matrix multiplication, i.e.,
X
(_ O OE)x,z:= _x,yO OEy,z.
y2Y
The category C(Y ; A) inherits in the obvious way the structure of an additive
category from A. We will often drop A from the notation.
If Y and A come with a G-action, we get a G-action on C(Y ; A) by (g*A)x :=
g*(Agx) and (g*OE)x,y:= g*(OEgx,gy). Here the action on Y is a left action, and*
* the
action on A is a right action, i.e., (g* O h*)(A) = (hg)*A. The action on C(Y ;*
* A) is
again a right action.
Denote by
CG (Y ; A)
the fixed point category. This is an additive subcategory of C(Y ; A). An objec*
*t in
CG (Y ; A) is given by a locally finite collection (Ax)x2Y of objects in A such*
* that
Ax = g*(Agx) holds for all g 2 G and x 2 Y . A morphism (OEx,y)x,y2Y in C(Y ; A)
between two objects which belong to CG (Y ; A) is a morphism in CG (Y ; A) if a*
*nd
only if g*(OEgx,gy) = OEx,yholds for all g 2 G and x, y 2 Y .
We are seeking certain additive subcategories of CG (Y, A), where support co*
*ndi-
tions are imposed on the objects and morphisms. This is formalized by the notion
of a coarse structure following [40]. For us it consists of a set E of subsets *
*of Y xY
and a set F of subsets of Y fulfilling certain axioms stated as (i) to (iv) in*
* [3,
page 167]. An object is called admissible if there exists F 2 F which contains *
*its
support. A morphisms (OEx,y) in CG (Y ; A) is called admissible if there exists*
* J 2 E
which contains its support supp(OE) := {(x, y) | x, y 2 Y, OEx,y6= 0}. The axio*
*ms are
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 9
designed such that the admissible objects together with the admissible morphisms
form an additive subcategory of CG (Y ; A) which we will denote by
CG (Y, E, F; A).
Let f :Y ! Z be a G-equivariant map. The formula (f*(A))z := y2f-1(z)Ay
defines a functor CG (Y, EY , FY ; A) ! CG (Z, EZ , FZ ; A) if f maps locally f*
*inite sets
to locally finite sets and takes EY to EZ and FY to FZ , see [3, Subsection 3*
*.3].
If g :Y ! Z is a second G-equivariant map that induces a functor, then there is
always a candidate for a natural equivalence between the two functors, namely we
can use the identity on each Ay. Viewed over Z this candidate for a morphism wi*
*ll
have a non-trivial support. This yields indeed a natural equivalence if the fol*
*lowing
holds.
(3.1) For each object A 2 CG (Y, EY , FY ; A) there is an element JA 2 EZ such
that (f(y), g(y)) 2 JA for all y 2 suppA.
3.2. Some control condition. Let Z be a space equipped with a quasi-metric d.
(We remind the reader that the difference between a metric and a quasi-metric is
that in the later case the distance 1 is allowed.) Then we define EZd to be the
collection of all subsets J of Zx Z of the form Jff= {(z, z0) | d(z, z0) ff} *
*with
ff < 1. A morphism ' 2 C(Z, EZd) is said to be ffi-controlled if supp' Jffi. *
*This
terminology will be used in subsection 6.3 and we will often be interested in s*
*mall
ffi.
Let Y be a G-space. A subset C Y is called G-compact if there exists a
compact subset C0 Y satisfying C = G . C0. For a G-CW -complex Y a subset
C Y is G-compact if and only if its image under the projection Y ! G\Y is a
compact subset of the quotient G\Y . Denote by FYGcthe set which consist of all
G-compact subsets of Y .
Let Y be a G-space. We denote by Gx the isotropy group of a point x 2 Y . Eq*
*uip
Y x[1, 1) with the G-action given by g(y, t) := (gy, t). As in [3, Definition 2*
*.7]) we
define EYGccto be the collection of subsets J (Y x[1, 1))x (Y x[1, 1)) satisf*
*ying
(3.2) For every x 2 Y , every Gx-invariant open neighborhood U of (x, 1) in
Y x[1, 1] there exists a Gx-invariant open neighborhood V U of (x, 1)
in Y x [1, 1] such that
((Y x[1, 1] - U)x V\)J = ;;
(3.3) The image of J under the projection (Y x[1, 1))x2 ! [1, 1)x2 sends J to
a member of E[1,1)dwhere d(t, s) = |t - s|;
(3.4) J is symmetric and invariant under the diagonal G-action.
EYGccis called the equivariant continuous control condition.
3.3. Controlled algebra and the assembly map. Let G be finitely generated
group equipped with a word-metric dG . For a G-space Y let p: Gx Y x[1, 1) !
Y x[1, 1), q :Gx Y x[1, 1) ! Gx Y and r :Gx Y x[1, 1) ! G be the canonical
projections. We will abuse notation and set
p-1EYGcc\ r-1EGdG:= {(px p)-1(J) \ (rx r)-1(J0) | J 2 EYGcc, J0 2 EGdG};
q-1FGxYGc := {q-1(F ) | F 2 FGxYGc}.
We define
T G(Y ; A):= CG (Gx Y, FGxYGc; A);
OG (Y ; A):= CG (Gx Y x[1, 1), p-1EYGcc\ r-1EGdG, q-1FYGxGc; A);
DG (Y ; A):= CG (Gx Y x[1, 1), p-1EYGcc\ r-1EGdG, q-1FYGxGc; A)1 .
10 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
We will often drop the A from the notation. Here the upper index 1 in the third
line denotes germs at infinity. This means that the objects of DG (Y ) are the *
*objects
of OG (Y ) but morphisms are identified if their difference can be factored ove*
*r an
object whose support is contained in Gx Y x[1, t] for some t 2 [1, 1), compare *
*[3,
Subsection 2.4].
We remark that in [3, Subsection 3.2] a slightly different definition of DG *
*(Y ) is
given, where the metric control condition EGdGdoes not appear. Using Theorem 3.7
below it can be shown that this does not change the K-theory of these categorie*
*s.
The metric control condition on G will be important in the construction of the
transfer, see in particular Proposition 6.13. The interested reader may compare*
* this
difference to different possible definitions of cone and suspension rings. Ofte*
*n it is
convenient to add finiteness condition to obtain formulas such as ( R)G = (RG),
compare [3, Remark 7.2].
The following is the so-called germs at infinity sequence.
(3.5) T G(Y ) ! OG (Y ) ! DG (Y ).
Here the first map is induced by {1} [1, 1) and the second is the quotient ma*
*p.
We will need the following facts.
Lemma 3.6.
(i)The sequence (3.5)induces a long exact sequence in K-theory;
(ii)The K-theory of OG (pt) is trivial.
Proof.We can replace T G(Y ) by an equivalent category, namely by the full subc*
*at-
egory of OG (Y ) on all objects that are isomorphic to an object in T G(Y ). Th*
*ese are
precisely the objects in OG (Y ) whose support is contained in Gx Y x[1, r] for*
* some
r 0. Then the first map in (3.5)becomes a Karoubi filtration and DG (Y ) is i*
*ts
quotient. Now (i)follows because Karoubi filtrations induce long exact sequences
in K-theory, see for example [16].
To prove (ii)it suffices to observe that there is an Eilenberg-swindle on DG*
* (pt)
induced by the map (g, t) 7! (g, t+1), compare for example [3, Proposition 4.4].
Theorem 3.7. The assignment Y 7! K*(DG (Y )) is a G-equivariant homology
theory on G-CW -complexes. The projection EF G ! ptinduces the assembly map
(0.2).
Proof.This is proven in [3, Section 5, Corrollary 6.3], see also [7, Theorem 7.*
*3]. As
mentioned above a slightly different definition is used in these references, bu*
*t this
does not affect the proof and the arguments can be carried over word for word.
The following is now an easy consequence, compare [37, Theorem 7.4].
Proposition 3.8. Suppose there exists an m0 2 Z such that for all A and all
m m0 we have
Km (OG (EF G; A)) = 0.
Then the assembly map (0.2)is an isomorphism for all n 2 Z and all A.
Proof.If the assembly map is an isomorphism for all m m0 and all A, then
it is an isomorphism for all n 2 Z and all A by [7, Corollary 4.7]. If we apply
Lemma 3.6 (i)to the map EF G ! ptwe obtain a map between homotopy fibration
sequences
K-1 T G(EF G) ____//_K-1 OG (EF G)____//K-1 DG (EF G)
| | |
| | |
|fflffl fflffl| fflffl|
K-1 T G(pt)________//K-1 OG (pt)______//K-1 DG (pt).
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 11
It is not hard to check that the left vertical map is induced by an equivalence
of categories and is therefore an equivalence of spectra. Because the homotopy
groups of the lower middle spectrum vanish by Lemma 3.6 (ii)the claim follows by
considering the long exact ladder of homotopy groups associated to the diagram
above.
3.4. The obstruction category as a functor of metric spaces. We will now
allow for (G, dG ) to be replaced by a metric space (Z, d) with a free G-action*
* by
isometries in the definition of OG (Y ; A)G . We define
OG (Y, Z, d; A) := CG (Zx Y x[1, 1), p-1EYGcc\ r-1EZd, q-1FZxGYc; A),
where p, q, r the same projections as before, but with G replaced by the free G-
space Z. As before we will often drop the A from the notation. The construction*
* is
functorial for G-equivariant maps f :Z ! Z0 that satisfy the following conditio*
*n.
(3.9) For every ff > 0 there exists a fi > 0 such that d(x, y) ff implies
d0(f(x), f(y)) fi.
Let (Zn, dn) be a sequence of metric spaces with free isometric G-action. We
define Y
OG (Y, (Zn, dn)n2N) OG (Y, Zn, dn)
n2N
as a subcategory of the indicated product category by requiring additional cond*
*i-
tions on the morphisms. A morphism ' = ('n)n2N is allowed if it is bounded with
respect to the sequence of metrics, i.e., if there exists a constant ff = ff(')*
*, such that
for every n 2 N and for every ((y,Lz, t), (y0, z0, t0)) 2 supp'n (Y xZnx [1, *
*1))x2
one has dn(z, z0) ff. The sum n2N OG (Y, Zn, dn) is in a canonical way a fu*
*ll
subcategory of OG (Y, (Zn, dn)n2N).
Later on, in Section 7, we will allow the dn to be quasi-metrics rather than
metrics. The definitions are clearly meaningful in this case as well.
These constructions are functorial for sequences of G-equivariant maps fn :Z*
*n !
Z0nthat satisfy the following uniform growth condition.
(3.10) For every ff > 0 there is fi > 0 such that for all n 2 N
dn(x, y) ff =) d0n(fn(x), fn(y)) fi.
4.The core of the proof
4.1. The map to the realization of the nerve. Let (Z, d) be a metric space.
Let U be a finite dimensional cover of Z by open sets.P Recall that points in t*
*he
realizationPof the nerve |U| are formal sums x = U2U xU U, with xU 2 [0, 1] s*
*uch
that U2U xU = 1 and such that the intersection of all the U with xU 6= 0 is
non-empty, i.e., {U | xU 6= 0} is a simplex in the nerve of U. There is a map
X
(4.1) f = fU :Z ! |U|, x 7! fU (x)U,
U2U
where
fU (x) = ___aU_(x)___P with aU (x) = d(x, Z - U) = inf{d(x, u) | u =2U}.
V 2UaV (x)
It is well-defined since U is finite dimensional. If Z is a G-space, d is G-inv*
*ariant
and U is an open F-cover, compare Definition 1.3, then the map f = fU is G-
equivariant. In our application fU will be strongly contracting with respect to*
* the
l1-metric on |U|, see Proposition 5.3.
12 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
4.2. The l1-metric on a simplicial complex. Every simplicial complex and in
particular the realization of the nerve of an open cover can be equippedPwith t*
*he
l1-metric,Pi.e., the metric where thePdistance between points x = U xU U and
y = U yU U is given by d1(x, y) = U |xU - yU |. We remark that this metric
does not generate the weak topology, unless the simplical complex is locally fi*
*nite.
We will never consider the weak topology and only be interested in the l1-metri*
*c.
__ __
4.3. The metric dC on Gx X . Let X be as in Theorem 1.1. We will now define_a
G-invariant_metric dC depending on a constant C > 0 on the G-space Gx X . Recall
that X is assumed_to be metrizable. We choose some (not necessarily G-invariant)
metric d__Xon X which generates the topology. We fix now for the rest of this p*
*aper
some choice of a word-metric dG on G.
__
Definition 4.2. Let C > 0. For (g, x), (h, y) 2 Gx X set
nX
dC ((g, x), (h, y)) = inf Cd__X(g-1ixi-1, g-1ixi) + dG (gi-1, gi),
i=1
where the infimum is taken over all finite sequences (g0, x0), (g1, x1), . .,.(*
*gn, xn)
with (g0, x0) = (g, x) and (gn, xn) = (h, y).
Proposition 4.3.
__
(i)dC defines a G-invariant metric on Gx X , with respect to the diagonal
action; __
(ii)dG (g, h) dC ((g, x), (h, y)) for all g, h 2 G and x, y_2_X ;
(iii)dG (g, h) = dC ((g, x), (h, x)) for all g, h 2 G and x 2 X ;
Proof.(i)It is immediate from the definition that dC is G-invariant, and satisf*
*ies the
triangle inequality. Because dG (g, h) 1 for all g 6= h we have dC ((g, x), (*
*h, y))
Cd__X(g-1x,_g-1y)} if g = h, and dC ((g, x), (h, y)) 1 if g 6= h, for all (g,*
* x), (h, y) 2
G x X. Hence dC is a metric.
(ii)and (iii)are obvious.
__ __
For C = 1 we will denote_the restriction of d1 to {g}x X = X by dg. Note that
considered as a metric on X this metric varies with g. Often we will be interes*
*ted
in de, where e denotes the unit element in G. (If the diameter of d__Xis less t*
*han
2, then de will in fact coincide with d__X, but this will not be important for *
*us.)_
Proposition 4.3 (i)implies that dg(x, y) = de(g-1x, g-1y) for g 2 G and x, y 2 *
*X .
__
4.4. The diagram. Let X be the G-space appearing in Theorem 1.1. Choose a
G-CW complex E which is a model for EF G, the classifying space for the family *
*F.
Fix an N 2 N as it appears_in Assumption 1.4 and for every n 2 N choose an open
F-cover U(n) of Gx X satisfying the conditions in Assumption 1.4 with_fi = n, i*
*.e.,
the dimension of U(n) is smaller than N and for every (g, x) 2 Gx X we can find
U 2 U(n) such that {g}nx {x} U. Here {g}n denotes the open ball with respect
to the word-metric dG in G of radius n around g. According to Lemma 5.1 below
we can choose for every n 2 N a constant C(n) such that the open F-cover U(n)
satisfies the following condition:
__
For every (g, x) 2 Gx X there exists a U 2 U(n) such that the
open ball of radius n with respect to the metric dC(n)around the
point (g, x) lies in U.
We will use the following sequences of metric spaces with free isometric G-acti*
*on
__ 1
(Gx X , dC(n))n2N, (Gx |U(n)|, dn)n2N.
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 13
Here the metric d1nis a product metric of the l1-metric on the simplicial compl*
*ex
|U(n)| scaled by the factor n and the word-metric dG on G, i.e.,
d1n((g, x), (h, y)) = dG (g, h) + nd1(x, y).
__
The map Gx X ! Gx |U(n)| defined by (g, x) 7! (g, fU(n)(g, x)) satisfies condi-
tion (3.9)and yields the functor
__ G 1
F U(n):OG (E, Gx X , dC(n)) ! O (E, Gx |U(n)|, dn).
We will construct the following diagram of additive categories around which the
proof is organized. Here the arrows labelled incare the obvious inclusions. T*
*he
functors pk and qk are defined by_first projecting onto the k-th factor and then
applying the projection map Gx X ! G and Gx |U(k)| ! G respectively. Both
projections clearly satisfy condition (3.9).
L G 1
(4.4) n2NO (E, Gx |U(n)|, dn)
(3)||
__ (2) fflffl|
OG (E,5(Gx5X , dC(n))n2N)```` ` `//OG (E, (Gx |U(n)|, d1n)n2N)
k
r n inc|| inc||
O Q fflffl|_ Qn2NFU(n) Q fflffl|
(1): n2N OG (E, Gx X , dC(n))________// n2NOG (E, Gx |U(n)|, d1n)
L P pk|| qk||
S fflffl| id fflffl|
OG (E)____________________________//OG (E)
The lower square commutes. In the remaining sections we will establish the foll*
*ow-
ing facts.
(4.5) After applying Km (-) for m 1 to the diagram the dotted arrow (1) exi*
*sts
and has the property that Km (pkOinc)O(1) is the identity on Km (OG (E))
for all k 2 N. This will be proven in Theorem 6.1; Q
(4.6) The dotted horizontal functor (2) defined as the restriction of n2NF *
*U(n)
to the indicated subcategories is well defined. This is the content of *
*Corol-
lary 5.6;
(4.7) The inclusion (3) from Subsection 3.4 gives an isomorphism on K-theory.
This follows from Theorem 7.2.
Proof of Theorem 1.1.According to Proposition 3.8 it suffices to show that the
group Km (OG (E)) vanishes for all m 1. So for m 1 apply Km to diagram (4.4*
*).
Pick an element
, 2 Km (OG (E))
at the lower left corner of the diagram. A quick diagram chase following the ar*
*rows
(1), (2) and (3) and using properties (4.5), (4.6)and (4.7)shows that there is
_ !
M
j 2 Km OG (E, Gx |U(n)|, d1n)
n2N
whose image under the map induced by qk O incO(3) is , for all k 2 N. Since
K-theory commutes with colimits (see Quillen [51, (12) on page 20]) we have the
canonical isomorphism
_ !
M ~= M
Km OG (E, Gx |U(n)|, d1n)-!Km OG (E, Gx |U(n)|, d1n).
n2N n2N
14 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
Hence there exists a k = k(j) 2 N such that for the projection prk onto the k-th
factor we get Km (prk)(j) = 0. This implies that the image of j under the map
induced by qk O incO(3) is trivial as well. This implies , = 0.
5. Contracting maps induced by wide covers
In this section we will use Assumption 1.4 to prove (4.6).
__
Lemma 5.1. Let fi 1. Suppose that U(fi) is an open F-cover_of Gx X as it
appears in Assumption 1.4, i.e., for every (g, x) 2 Gx X there exists U 2 U(fi)
such that {g}fix{x} U. Then there exists a constant C = C(U(fi)) > 1 such that
the following holds: __
For every (g, x) 2 Gx X there exists U 2 U(fi) such that the open
fi-ball with respect to the metric dC around (g, x) lies in U.
__
Proof.For every z 2 X we can find by assumption Uz 2 U(fi) with {e}fix{z} Uz,
where e_2_G is the unit element. Choose for gT2 {e}fian open neighborhood Vg,z_
of z 2 X such that {g} x Vg,z Uz. Put Vz := _g2{e}fiVg,z._Then {Vz | z 2 X }
is an open cover of the compact metric space_(X , d__X). Let " > 0 be a Lebesgue
number for_this open cover, i.e., for x 2 X the ball x" lies in Vz(x)for an app*
*ropriate
z(x) 2 X ._ __ __
Since X is compact, the map X ! X , x 7! gx is uniformly continuous. Hence_
we can find ffi(", g) > 0 such that d__X(gx, gy) < _"fiholds for all x, y 2 X *
*with
d__X(x, y) < ffi(", g). Since there are only finitely many elements in {e}fi, *
*we can
choose a constant C such that fi_C< ffi(", g) holds for all g 2 {e}fi. Thus we *
*get
__ fi fi
(5.2) d__X(gx, gy) < "_fiforx, y 2 X with d__X(x, y) < __Candg 2 {e} .
Because dC and the cover U are G-invariant,_it suffices to prove the claim f*
*or an
element of the shape (e, x) 2 G x X. Let (h, y) be an element in the ball of ra*
*dius
fi around (e, x) with respect to the metric dC . We want to show (h, y) 2 Uz(x)*
*. By
definition of dC we can find a sequence_of_elements (e, x) = (g0, x0), (g1, x1)*
*, . .,.
(gn-1, xn-1), (gn, xn) = (h, y) in G x X such that
Xn Xn
dG (gi-1, gi) + C . d__X(g-1ixi-1, g-1ixi) < fi.
i=1 i=1
We can arrange gi-16= gi, otherwise delete the element (gi, xi) from the sequen*
*ce,
the inequality above remains true because of the triangle inequality for d__X. *
*Since
dG (gi-1, gi) 1, we conclude
n fi.
By the triangle inequality dG (e, gi) fi for i = 1, 2, . .,.n. In other words*
* gi2 {e}fi
for i = 1, 2, . .,.n.
We have d__X(g-1ixi-1, g-1ixi) < fi_Cfor i = 1, 2, . .,.n. We conclude from*
* (5.2)
that "
d__X(xi-1, xi) < __fi
holds for i = 1, 2, . .,.n. The triangle inequality implies together with n fi
d__X(x, y) < ".
Hence y 2 Vz(x). Since h 2 {e}fiholds, we conclude y 2 Vz(x) Vh,z(x). This
implies (h, y) 2 Uz(x).
The following proposition yields contracting properties of the map from a me*
*tric
space to the nerve of an open cover of the space. Similar ideas appear already *
*in
Section 1 of [35].
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 15
Proposition 5.3. Let X = (X, d) be a metric space and let fi 1. Suppose U is *
*an
open cover of X of dimension less than or equal to N with the following propert*
*y:
For every x 2 X there exists U 2 U such that the fi-ball around
x lies in U.
Then the map fU :X ! |U| (defined in Subsection 4.1) has the following contract*
*ing
property. If d(x, y) _fi_4Nthen
2
d1(fU (x), fU (y)) 16N__fid(x, y).
Note that if fi gets bigger, the estimate applies more often and fU contrac*
*ts
stronger.
P aU(x)
Proof. Recall that fU (x) = U fU (x)U, where fU (x) = ________PVaV(x)with aU *
*(x) =
d(x, X - U) = inf{d(x, u) | u =2U}. For every V 2 U we set bV (x, y) = aV (x) -
aV (y). Since d is a metric we have |bV (x, y)| d(x, y). Since the covering d*
*imension
is smaller than N there are at most 2N covering sets V for which bV (x, y) 6= 0.
Hence we have
X fi
(5.4) |bV (x, y)| 2Nd(x, y) __.
V 2
For every x there exists by assumption U 2 U such that the fi-ball around x is
contained in U. For this U we have
X
(5.5) aV (x) aU (x) fi.
V
We compute
P P
aU (x) VbV (x, y) - bU (x, y) VaV (x)
fU (y) - fU (x) = ___________________________________(P. P
V aV (x))( *
*V aV (x) - bV (x, y))
Now one can estimate using (5.4)for the third, (5.4)and (5.5)for the fourth
inequality and (5.4)for the last inequality.
X X fifiP bV (x, y) fifi X fifi bU (x, y) fifi
|fU (x) - fU (y)| fifi__V_____________Pfifi+ fifi________________Pfi*
*fi
U U VPaV (x) - bV (x, y) U V aV (x) - bV (x, y)
|bV (x, y)|
4N ______V_____________| P
VaV (x) - bV (x, y)|
4N _____2Nd(x,_y)______| P
VaV (x) - bV (x, y)|
2d(x, y)
______8N______________PP
V aV (x) - |bV (x, y)|
2d(x, y) 8N2d(x, y) 16N2d(x, y)
_8N_________fi -_2Nd(x,_y)_fi= ___________.
fi - _2 fi
Combining these two statements we can now establish (4.6).
Corollary 5.6. The map (2) in diagram (4.4)is well defined.
Proof.Let ' = ('n) be a morphism in the source, then there exists a constant K =
K(')_such_that for every n 2 N we have that ((g, x, e, t), (g0, x0, e0, t0)) 2 *
*supp'n
(Gx X xEx [1, 1))x2 implies dC(n)((g, x), (g0, x0)) K. By Proposition 4.3 (ii*
*)it
suffices to show that there exists a constant L such that
nd1(fU(n)((g, x)), fU(n)((g0, x0))) L,
16 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
compare (3.10). By the construction of the sequence (C(n))n2N_the assumptions
in Proposition 5.3 are satisfied for the cover U(n) of Gx X with fi = n for eve*
*ry
n 2 N. We conclude for n 4KN and ((g, x, e, t), (g0, x0, e0, t0)) 2 supp'n th*
*at
nd1(fU(n)((g, x)), fU(n)((g0, x0))) 16N2dC(n)((g, x), (g0, x0)) 16N2K =:*
* L.
The distance of two points of a simplicial complex with respect to the l1-metri*
*c is
at most 2. Because (4KN) . 2 L this implies that the above estimate holds in
fact for all n.
6.The transfer
In this section we will use Assumption 1.2 to deal with the dotted arrow (1)*
* in
diagram (4.4). The following result establishes (4.5).
Theorem 6.1. Let m 1. There exists a map
__
trans:Km OG (E) ! Km OG (E, (Gx X , dC(n))n2N)
that is for all k 2 N a right inverse for the map induced by prk:= pk O inc, co*
*mpare
(4.4).
This will be proven as follows. For an additive category O of the type appea*
*ring
above we define Waldhausen categories chhfdO and echhfdO together with natural
inclusions
O inc--!chhfdO inc--!echhfdO
that induce isomorphisms on Km for every m 1, compare Lemma 6.5. We then
construct the functor transin Subsection 6.4 (see in particular Proposition 6.1*
*3)
in order to obtain for every k 2 N the following diagram of Waldhausen categori*
*es
and exact functors
(6.2) echhfdOG (E, (Gx __X, dC (n))n2N)
kk55k
transkkkkkkk |echhfd(prk)|
kkkkk fflffl|
OG (E)______inc____//echhfdOG (E).
In Lemma 6.16 we show that this diagram commutes after applying K-theory. From
this Theorem 6.1 follows.
6.1. Review of the classical transfer. As a motivation for the forthcoming
construction we briefly review the transfer for the Whitehead group associated *
*to
a fibration F ! E -p!B of connected finite CW -complexes. Recall that the fiber
transport gives a homomorphism t: ss1(B) ! [F, F ]. Under mild conditions on t
one can define geometrically a transfer homomorphism trans:Wh (B) ! Wh (E)
by sending the Whitehead torsion of a homotopy equivalence_f :B0 ! B of finite
CW -complexes to the Whitehead torsion of the pull back f:p*E ! E (see [1], [43,
Section 5]). An algebraic description in terms of chain complexes is given in [*
*43,
Section 4]) and is identified with the geometric construction. It involves the *
*chain
complex of an appropriate cover of F and the action up to homotopy of ss1(B)
coming from the fiber transport. The map trans:Wh (B) ! Wh (E) is bijective if
F is contractible.
The desired transfer
__
trans:OG (E) ! echhfdOG (E, (Gx X , dC(n))n2N)
is a controlled version on the category level of_the_algebraic description of t*
*he
classical transfer above in the situation Gx Ex X ! Gx E which one may consider
after_dividing out the diagonal G-action as a flat bundle with the contractible*
* space
X as fiber and (Gx E)=G ~=E as base space. The group G plays the role of ss1(B)
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 17
__
and the fiber transport comes from the honest G-action on X . Having this in mi*
*nd
it becomes clear why in the sequel we will have to deal with categories of chain
complexes.
6.2. Some preparations. Fix an infinite cardinal ~. Let F~(Z) be a small model
for the category of all free Z-modules which admit a basis B with card(B) ~.
Let Ff(Z) be the full subcategory of F~(Z) that consists of finitely generated *
*free
Z-modules. These categories will always be equipped with the trivial G-action. *
*Let
A be an additive category with G-action. According to [7, Lemma 9.2] there exist
additive categories with G-action Af and A~ together with G-equivariant inclusi*
*on
functors
A ! Af ! A~.
In A~ there exist direct sums with indexing sets of cardinality than or equal t*
*o ~
and A ! Af is an equivalence of categories. There exists a "tensor product", i.*
*e.,
a bilinear functor
(6.3) - -: A~x F~(Z) ! A~
which is compatible with the G-action on A~, i.e., g*(A M) = g*A M and
restricts to
- -: Afx Ff(Z) ! Af.
For all practical purposes we can and will identify A with Af.
For a G-space Y and a metric space (Z, d) with a free action of G by isometr*
*ies
we define the category __
O G(Y, Z, d; A~)
in the same way as before but we replace A by A~ and drop the assumption that
the support of objects is locally finite. Moreover, instead of defining a morph*
*ism
' to be a family of morphisms 'y,xand requiring that for fixed x, respectively
y, the sets {y | 'y,x 6= 0}, respectively {x | 'y,x 6= 0}, are finite we define*
* a
morphism ': A = (Ax) ! B = (By) to be a morphism xAx ! yBy in the
category A~. Note that for suitably chosen ~ these direct sums exist in A~, com*
*pare
[7, Lemma 9.2]. For a sequence (Zn, dn)n2N of metric spaces with G-action by
isometries we define
__G ~ Y __G ~
O (Y, (Zn, dn)n2N; A ) O (Y, Zn, dn; A )
n2N
by requiring conditions on the morphisms precisely as in Subsection 3.4. One sh*
*ould
think of the inclusions
__G ~
(6.4) OG (Y ; A) O (Y ; A ),
__G ~
OG (Y, (Zn, dn)n2N; A) O (Y, (Zn, dn)n2N; A ), etc.
as inclusions of full subcategories on objects satisfying finiteness conditions*
* into
large categories which give room for constructions. The prototype of such a sit*
*ua-
tion is the inclusion Ff(Z) F~(Z).
Let O be an additive category. We write Idem(O) for its idempotent completio*
*n.
We define chfO to be the category of chain complexes in O that are bounded above
and below and ch O to be the category of chain complexes that are bounded
below. For these categories the notion of chain homotopy leads to a notion of w*
*eak
equivalence, and we define cofibrations to be those chain maps which are degree*
*wise
the inclusion_of a direct summand. __
Now let O be an additive category and let O O be the inclusion of a full
additive subcategory. We write
__
chhf(Idem (O) Idem(O ))
18 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
__
for the full subcategory of ch Idem (O ) consisting of chain complexes that are*
* chain
homotopy equivalent to a chain complex in chfIdem(O). We write
__
chhfd(O Idem(O ))
__
for the full subcategory of ch Idem (O ) consisting of objects C which are homo*
*topy
retracts of objects in chfO, i.e., there exists a diagram C -i!D -r!C with D in
chfO such that the composition r O i is chain homotopic to the identity on C.
Lemma 6.5. We have an equality
__ __
chhf(Idem (O) Idem(O )) = chhfd(O Idem(O ))
and the inclusions
__
O __________//_chfO_________//_chhfd(O Idem(O ))
| | ||
| | ||
fflffl| fflffl| || __
Idem(O) ____//_chfIdem(O)____//chhf(Idem (O) Idem(O ))
induce equivalences on Km for all m 1.
Proof.Suppose C is a chain complex in chfIdem(O). Then by adding elementary
chain complexes of the form
. .!.0 ! 0 ! P -id!P ! 0 ! 0 ! . . .
one can produce a chain homotopy equivalent chain complex C0 in Idem(O) such
that all C0iexcept the one in the top-non-vanishing dimension n lie in O instea*
*d of
Idem (O). By adding a complement to C0none can easily produce a chain complex
in chfO which contains C0 as an (honest) retract. Since the homotopy relation is
transitive this proves the inclusion " ". __
Suppose we have C -i!D r-!C with r O i ' id, where C lies in ch Idem (O ) and
D in chfO. Then the proof of Proposition 11.11 in [44] yields a chain complex D0
in ch O which is chain homotopic to C and of a special form. Namely, there exis*
*ts
an n 2 Z and an object D01 such that D0m= D01 for all m n. Moreover there
exists a map p: D01! D01with p O p = p such that the chain complex is 2-periodic
above n and of the form
. .1.-p--!D01-p!D01-1-p-!D01-p!D01= D0n! D0n-1! D0n-2! . ...
In ch Idem (O) this chain complex is homotopic to
. .!.0 ! 0 ! (D01, p) p-!(D0n, id) ! (D0n-1, id) ! (D0n-2, id) ! . ...
This proves the other inclusion. The two horizontal inclusions on the left are *
*well
known to induce isomorphisms on Km , for m 0, compare [56], [16]. The lower
horizontal inclusion on the right induces an isomorphism on Km , for m 0 by
an application of the Approximation Theorem 1.6.7 in [57]. The vertical inclusi*
*on
on the left induces an isomorphism on Km for m 1 by the Cofinality Theorem,
compare Theorem 2.1 in [55].
Notation 6.6. In the following we will use the abbreviation
__
chhfdO = chhfd(O Idem(O ))
__
because O will be clear from the context. In fact we will always be in the situ*
*ation
described in (6.4).
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 19
We recall from [8, Subsection 8.2] that for a given Waldhausen category W th*
*ere
exists a Waldhausen category fW whose objects are sequences
C0 c0-!C1 c1-!C2 c2-!. .,.
where the cn are morphisms in W that are simultaneously cofibrations and weak
equivalences. A morphism f in fW is represented by a sequence of morphisms
(fm , fm+1 , fm+2 , . .).which makes the diagram
cm+1 cm+2
Cm __cm___//_Cm+1______//_Cm+2_____//. . .
fm|| fm+1|| fm+2||
fflffl|dm+k fflffl|dm+k+1fflffl|dm+k+2
Dm+k _____//Dm+k+1_____//Dm+k+2_____//. . .
commutative. Here m and k are non-negative integers. If we enlarge m or k the r*
*e-
sulting diagrams represent the same morphism, i.e., we identify (fm , fm+1 , fm*
*+2 , . .).
with (fm+1 , fm+2 , fm+3 , . .).but also with (dm+k O fm , dm+k+1 O fm+1 , dm+k*
*+2 O
fm+2 , . .).. Sending an object to the constant sequence defines an inclusion
W ! fW.
According to [8, Proposition 8.2] the inclusion induces an isomorphism on Km for
m 0 under some mild conditions about W. These conditions will be satisfied in
all our examples.
__
6.3. The singular chain complex of X . In the next subsection we will construct
the functor denoted_transin diagram (6.2). It will essentially replace objects *
*A 2 A
by A Csing,ffi*(X , de). Here we use a chain_subcomplex of the singular chain c*
*omplex
and consider it as a chain complex over X . We now collect some facts about the
singular chain complex of a metric space that will be needed in the constructio*
*n of
the transfer.
Let X = (X, d) be a metric space. As before we denote the singular chain
complex of X by Csing*(X). For ffi > 0 we define
Csing,ffi*(X, d) Csing*(X)
as the chain subcomplex generated by all singular simplices oe : ! X for which*
* the
diameter of oe( ) is less or equal to ffi, i.e., for all y, z 2 we have d(oe(*
*y), oe(z)) ffi.
This chain complex can be considered as a chain complex over X via the baryc*
*en-
ter map, i.e., for x 2 X the module Csing,ffin(X, d)x is generated by all singu*
*lar n-
simplices which satisfy the condition above and map the barycenter to x. A map
f :C* ! D* of chain complexes over X is called a ffi-controlled homotopy equiva-
lence if there exists a chain homotopy inverse g and chain homotopies h: f O g *
*' id
and h0:g O f ' idsuch that f, g, h and h0are all ffi-controlled when considered*
* as
morphisms over X, see Subsection 3.2.
Lemma 6.7. Let X = (X, d) be a metric space.
(i)For ffi0> ffi > 0 the inclusion
0
i: Csing,ffi*(X, d) ! Csing,ffi*(X, d)
is a ffi0-controlled chain homotopy equivalence;
(ii)For every ffi > 0 the inclusion
i: Csing,ffi*(X, d) ! Csing*(X)
is a chain homotopy equivalence;
20 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
(iii)Suppose X = |T | is a simplicial complex, i.e., the realization of an a*
*bstract
simplicial complex T . Let C*(T ) denote the simplicial chain complex c*
*on-
sidered as a chain complex over X = |T | via the barycenters. Suppose a*
*ll
simplices of |T | have diameter smaller than ffi. Then realization defi*
*nes a
map
C*(T ) ! Csing,ffi*(X, d)
which is a ffi-controlled chain homotopy equivalence.
Proof.(i)Let C denote the category whose objects are the closed subsets of X and
whose morphisms are the inclusions. We can consider
X A 7! Csing,ffi*(A) = Csing,ffi*(A, d|A )
as a functor from C to the category of chain complexes, i.e., as a ZC-chain com*
*plex.
(For basic definitions and facts of ZC-modules we refer0to [44, Section 9].) We
claim that the inclusion Csing,ffi*(?) ! Csing,ffi*(?) is a chain homotopy equi*
*valence of
ZC-chain complexes. Note that for every n 2 Z
M
Csing,ffin(?) = Z morC(oe( ), ?)
oe
is a free ZC-chain complex, here the sum runs over all singular simplices in X *
*whose
image have a diameter less or equal to ffi. Because of the fundamental theorem *
*for
homological algebra in the setting of RC-chain complexes (see [44, Lemma 11.7 on
page 220]), it suffices to prove that for every closed subset A X the inclusi*
*on
0
(6.8) Csing,ffi*(A) ! Csing,ffi*(A)
induces a homology isomorphism. In order to see this one uses that the usual
subdivision chain0selfmap sd of the singular chain complex restricts to a selfm*
*ap
of Csing,ffi(A) and so does the chain homotopy proving that sdis homotopic0to t*
*he
identity. Moreover for each individual singular simplex oe in Csing,ffi*(A) the*
*re exists
an N, such that sdN oe lies in Csing,ffi(A) by a Lebesgue-number argument.
We now have a homotopy inverse p and homotopies h and h0as maps of ZC-chain
complexes. Evaluating p at X yields a chain homotopy inverse pX of ordinary cha*
*in
complexes that restricts to every0closed subset of X. In particular for every s*
*ingular
simplex oe : ! X in Csing,ffi*(X, d) the image under pX lies in Csing,ffi*(oe*
*( ), d).
Hence pX considered as a morphism over X is bounded by ffi0 because oe( ) lies
within a ffi0-ball around oe(bary( )) and the same argument works for the homo-
topies hX and h0X.
(ii)It suffices to prove that the inclusion induces a homology isomorphism. *
*This
is a less careful version of the argument used above for the map (6.8).
(iii)The proof starts similar to the proof of assertion (i). Instead of the *
*cate-
gory C of closed subsets and inclusions one works with the category of simplici*
*al
subcomplexes of T and inclusions. Let S T be a simplicial subcomplex then the
composition in the sequence
C*(S) ! Csing,ffi*(|S|) ! Csing*(|S|)
is well known to be a homology isomorphism and the second map is a homology
isomorphism by assertion (ii). If we evaluate at S = T we see that the map
in question is a homotopy equivalence and that the homotopy inverse and the
homotopies can be chosen in such a way that they restrict to every simplex. Sin*
*ce
the simplices have diameter at most ffi we see that these maps are ffi-controll*
*ed.
__
Next we prove_for_X as in Theorem 1.1 and the metric de from Subsection 4.3
that Csing,ffi*(X , de) is homotopy finitely dominated in a controlled sense. W*
*e will
make use of Assumption 1.2, i.e., we assume the following.
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 21
__ __
There exists_a homotopy H :X x[0, 1] ! X , such that H0 = id_X
and Ht(X ) X for every t > 0.
*
* __
Lemma 6.9. Let ffi > 0 be given. There exists a finite chain complex Dffi*in ch*
*fC(X )
all whose differentials are ffi-controlled with respect to de together with maps
__ i ffir sing,ffi_
Csing,ffi*(X ) -!D* -!C* (X )
and a chain homotopy_h:_r O i ' id such that i, r and h are bounded by 6ffi as
morphisms over X = (X , de).
Proof.Let H be a homotopy as in Assumption_1.2. For t_>_0 let Kt be the union
of all simplices of X that meet Ht(X_) X. Since Ht(X ) is compact this is a f*
*inite
simplicial subcomplex of X. Since X xI is compact H is uniformly continuous.
Since H0 is the identity, we can find for a given_ffi > 0 an " = "(ffi) > 0 suc*
*h that
H({x}"x [0, "]) {x}ffi=2holds_for all x 2 X . We conclude that H({x}ffix[0, "*
*]) __
{x}2ffiholds for x 2 X . In particular H" maps ffi-balls to 2ffi-balls. Let inc*
*:K" ! X
be the inclusion. Then
__ (H")* sing,2ffi inc* sing,2ffi_
Csing,ffi*(X ) ----! C* (K") --! C* (X )
is well defined and the composition is homotopic to the inclusion
__ inc* sing,2ffi_
Csing,ffi*(X ) --! C* (X )
via a homotopy that is 2ffi-controlled. The latter inclusion is a 2ffi-control*
*led ho-
motopy equivalence by Lemma 6.7 (i). After a suitable subdivision we can assume
that in the simplicial complex K" = |T"| all simplices have diameter smaller th*
*an
ffi. Lemma 6.7 (iii)says that there exists a 2ffi-controlled homotopy equivale*
*nce
C(T") ! Csing,2ffi*(K"). Now set Dffi*= C(T").
6.4. The controlled transfer. Fix an infinite cardinal ~ large enough such that
the_following_constructions make sense. For ffi > 0 we define a chain complex o*
*ver
Gx X , more precisely a chain complex
__G __ ~
C*(ffi) 2 ch C (Gx X ; F (Z))
__
as follows. The n-th module Cn(ffi) is as a module over Gx X given by
__
(6.10) Cn(ffi)(g,x)= Csing,ffin(X , de)g-1x.
(Note that Cn(ffi) is indeed an object in_the_subcategory that is fixed under G*
*.)
Here de is the (non-invariant) metric on X from Subsection 4.3. The differenti*
*al
@ :Cn(ffi) ! Cn-1(ffi) is given by
ae 0
@(g0,x0),(g,x)= @g-1x0,g-1x0ifgo=tgherwise,
where @g-1x0,g-1xare the components of the differential
__ sing,ffi_
@ :Csing,ffin(X , de) ! Cn-1 (X , de),
__
considered_as a map over X . Note that differentials have non-diagonal support *
*only
in the X -direction.
Similarly using the chain complexes_Dffi*appearing in Lemma 6.9 we define a
chain complex D*(ffi) over Gx X by
D*(ffi)(g,x)= (Dffi*)g-1x.
22 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
Lemma 6.11. Let ffi > 0 and C > 1. The chain complex C*(ffi) is a homotopy
retract of the chain complex D*(ffi). The differentials of C*(ffi) and D*(ffi) *
*and the
maps and homotopies proving that C*(ffi) is a homotopy retract satisfy the foll*
*owing
control condition. If ((g0, x0), (g, x)) lies in the support of one of these ma*
*ps, then
g0= g and dC ((g, x0), (g, x)) 6Cffi.
Proof. Note_that_C*(ffi) is the unique G-invariant_chain complex whose restrict*
*ion
to {e}x_X Gx X coincides with Csing,ffi*(X_,_de) considered as a chain comple*
*x over
{e}x X via the identification {e}x X ~=X . Similarly_one can extend all the maps
and homotopies from Lemma 6.9 to maps over Gx X . The statement about the
support of these maps follows immediately from the definitions.
Now let (C(n))n2N be the sequence of numbers C(n) > 1 that we have chosen
towards the beginning of Subsection 4.4. Assume that (ffi(n))n2N is a sequence *
*of
positive numbers which satisfies the following condition.
(6.12) There exists a constant ff > 1 such that ffi(n) _ff_C(n)for all n 2 N.
Depending on the sequence (ffi(n))n2N we now would like to define the transf*
*er
functor __
trans:OG (E) ! chhfdOG (E, (Gx X , dC(n))n2N).
However, we will see soon that we have to modify the target category in order to
get a well defined functor. In order to motivate this modification we describe *
*the
naive attempt to define the functor and explain where the problem occurs. On
objects the functor should be given by
A 7! (A C*(ffi(n)))n2N,
__
where A Ck(ffi(n)) is an object over Gx X xEx [1, 1) via
*
* __
(A Ck(ffi(n)))(g,x,e,t)= A(g,e,t) Ck(ffi(n))(g,x,t)= A(g,e,t) Csing,ffi(n)k*
*(X , de)g-1x,
and the differentials are given by
ae 0 0 0
(id @(n))(g0,x0,e0,t0),(g,x,e,t)= id0@(g0,x0),(g,x)if(go,teh,etr)w=i(g,se,*
*et);.
__
Again off-diagonal support for the differentials occurs only in the X -directio*
*n.
Lemma 6.11 and (6.12)imply_that (A C*(ffi(n)))n2N is a well defined object in
chhfdOG (E, (Gx X , dC(n))n2N).
On morphisms a problem occurs. We would like to map the morphism ': A ! B
with components '(g0,e0,t0),(g,e,t):A(g,e,t)! B(g0,e0,t0)to the morphism (' l(n*
*))n2N
whose components are given by
(' l(n))(g0,x0,e0,t0),(g,x,e,t)= '(g0,e0,t0),(g,e,t) l(n)(g0,x0),(*
*g,x),
with ae
0= x;
l(n)(g0,x0),(g,x)= l(n)g0-1g0 ifxifx06= x,
where l(n)g0-1gis the map
__ sing,ffi(n)_
(lg0-1g)*: Csing,ffi(n)k(X , de)g-1x! Ck (X , de)g0-1x
*
* __
which is induced by left multiplication_with_g0-1g, i.e., a singular simplex oe*
* : ! X
is mapped to lg0-1gO oe, where lg0-1g:X ! X is the map x 7! g0-1gx. However the
map l(n)g0-1gis not well defined, its target is not as stated. One only has a w*
*ell
defined map (in fact an isomorphism)
__ sing,ffi(n)_
(lg0-1g)*: Csing,ffi(n)k(X , de)g-1x! Ck (X , dg0-1g)g0-1x,
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 23
where in the target we work with the metric dg0-1ginstead of de. We will compose
this map with the inclusion
__ sing,effi(n)_
Csing,ffi(n)k(X , dg0-1g)g0-1x Ck (X , de)g0-1x
for a suitable chosen (effi(n))n2N with effi(n) ffi(n) in order to at least o*
*btain a well
defined morphism
(' l(n))n2N :(A C*(ffi(n)))n2N ! (B C*(effi(n)))n2N,
Now the f -construction that was reviewed at the end of Subsection 6.2 comes in*
*to
play.
Proposition 6.13. Choose a collection of numbers ffik(n), k 2 N, n 2 N as in
Lemma 6.14. Then there exists a functor depending on that choice
__
trans:OG (E) ! echhfdOG (E, (Gx X , dC(n))n2N)
which sends a morphism ': A ! B to the morphism whose n-th component is
represented by
A C*(ffiff(n))id_/inc/A C*(ffiff+1(n))id//inc_A C*(ffiff+2(n))id//inc*
*_. . .
'|l(n)| |'|l(n) '|l(n)|
fflffl|id inc fflffl| id inc fflffl|id inc
B C*(ffiff+1(n))__//_B C*(ffiff+2(n))//_B C*(ffiff+3(n))//_... .
Here ff = ff(') 2 N is chosen such that for every ((g0, e0, t0), (g, e, t)) s*
*upp' we
have dG (g, g0) ff.
It is here that we use the metric control condition on G in the definition of
OG (E): it ensures the existence of ff in the above statement.
Proof.The boundedness condition with respect to the sequence of metrics (dC(n))*
*n2N
for the differentials follows because of Lemma 6.14 (ii). That we have a homoto*
*py
finitely dominated object follows from Lemma 6.11. Hence (A C*(ffik(n)))n2N is
indeed a well defined object. Lemma 6.14 (i)assures that one has the horizontal
inclusions. The vertical maps exist because of Lemma 6.14 (iii). Because the E-
coordinate is left unchanged in this construction, the equivariant continuous c*
*ontrol
condition is preserved.
Lemma 6.14. Let (C(n))n2N be a monotone increasing sequence of numbers.
There exists a collection of numbers ffik(n) > 0 with n, k 2 N, such that the f*
*ollowing
conditions are satisfied.
(i)For every fixed n 2 N the sequence (ffik(n))k2N is increasing, i.e.,
ffi1(n) ffi2(n) ffi3(n) . .;.
(ii)For every k 2 N there exists ff(k) 0 such that
ffik(n) ff(k)_C(n)
for all n 2 N; __
(iii)Consider g, h 2 G, x, y 2 X and k, n 2 N. If dG (g, h) k and dg(x, y)
ffik(n), then
dh(x, y) ffik+1(n).
24 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
Proof. For L 2 N and ffi 0 put
eRL(ffi):= sup{dg(x, y) |(g, x, y) 2 Gx __Xx_Xwith dG (g, e) L, de(x, y) f*
*fi};
RL(ffi):= max {ffi, eRL(ffi)}.
__
Since X is compact and there are only finitely many g with dG (g, e) L, this *
*defines
a monotone increasing map RL :[0, 1) ! [0, 1) with RL(ffi) ffi. In particular
RL(ffi) > 0 for ffi > 0. Moreover_RL_is_continous at 0 because the identity yie*
*lds a
uniformly continous map (X , de) ! (X , dg). Note that RL(ffi) RL+1(ffi). Usi*
*ng
dg(x, y) = de(g-1x, g-1y) we can conclude that
(6.15) dG (g, h) L and dg(x, y) ffi impliesdh(x, y) RL(ffi).
Define R-1L(ffi) = min{ffi, sup{ff 2 [0, 1) | RL(2ff) ffi}}. Here by abuse of*
* notation
R-1Lstands for some function but need not be the inverse of RL. The function
R-1Lis monotone increasing and satisfies 0 < R-1L(ffi) ffi for ffi > 0. We c*
*laim
that RL(R-1L(ffi)) ffi. In fact for s < R-1L(ffi) we have RL(2s) ffi. Hen*
*ce for
s = 3_4R-1L(ffi) we have by monotony RL(R-1L(ffi)) RL(23_4R-1L(ffi)) ffi. F*
*or n 2 N
define
ffin(n) = __1__C(n).
For k = n + l with l 1 put
ffik(n) = Rn+l-1O . .O.Rn+1 O Rn(ffin(n))
and for k = n - l, with l = 1, 2, . .,.n - 1 set
ffik(n) = R-1n-lO . .O.R-1n-2O R-1n-1(ffin(n)).
It remains to check that the numbers ffik(n) have the desired properties.
(i)This follows since RL(ffi) ffi and R-1L(ffi) ffi.
(ii)For n k we have ffik(n) ffin(n) = _1__C(n)by (i). Now we can choose ff(*
*k) to be
max {1, C(n)ffik(1), . .,.C(n)ffik(k - 1)}.
(iii)Since Rk(R-1k(ffi)) ffi we conclude
Rk(ffik(n)) ffik+1(n).
We derive from (6.15)
dh(x, y) Rk(ffik(n)) ffik+1(n).
Lemma 6.16. After applying K-theory diagram (6.2)is commutative.
Proof.Because of [57, Proposition 1.3.1] it suffices to construct a natural tra*
*ns-
formation T of functors OG (E) ! echhfdOG (E) between echhfd(prk) O transand the
obvious inclusion such that T (A) is a weak homotopy equivalence in echhfdOG (E)
for every object A in OG (E).
Consider a ZG-chain complex C* such that after forgetting the G-action
C* 2 chhfdFf(Z) = chhfd(Ff(Z) F~(Z)).
__ sing,ffi_
Examples are Csing*(X ) and C* (X ) by Lemma 6.7 (ii)and the (easier) uncon-
trolled version of Lemma 6.9. We define a functor
lC* :OG (E) ! chhfdOG (E)
as follows. Let A = (A(g,y,t))(g,y,t)2GxEx[1,1)and B = (B(g0,y0,t0))(g0,y0,t0)2*
*GxEx[1,1)
be objects in OG (E) and let ': A ! B be a morphism in OG (E) with components
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 25
'(g0,y0,t0),(g,y,t):A(g,y,t)! B(g0,y0,t0). Define A 7! A C* and ' 7! ' l, w*
*here for
(g, y, t), (g0, y0, t0) 2 G x E x [1, 1) we put
(A C*)(g,y,t)= A(g,y,t) C*
with differential
@(g0,y0,t0),(g,y,t)= id(g0,y0,t0),(g,y,t) @
and
(' l)(g0,y0,t0),(g,y,t)= '(g0,y0,t0),(g,y,t) lg0-1g,
where lg0-1gis left multiplication with g0-1g.
Let C* and D* be ZG-chain complexes belonging to chhfdFf(Z) and f*: C* !
D* be a Z-chain map. Then for every object A in OG (E), we have an induced
chain map idA f*: lC*(A) ! lD*(A). If f* is moreover a ZG-chain map, then this
construction is compatible with lC* and lD* on morphisms and defines a natural
transformation lf*:lC* ! lD* of functors. If f* is a chain homology equivalence*
* (af-
ter forgetting the group action), then lf*(A): lC*(A) ! lD*(A) is a chain homot*
*opy
equivalence in chhfdOG (E): since C* and D* are free as Z-chain complexes, we c*
*an
choose a (not necessarily G-equivariant) Z-chain homotopy inverse u*: D* ! C*
for f*. Then idA u* is a homotopy inverse for idA f* = lf*(A).
Let 0(Z)* be the ZG-chain complex concentrated in dimension_zero and given
there by Z with the trivial G-operation. Let "*: Csing*(X ) ! 0(Z)*nbe_the_ZG-c*
*hain
complex map given by augmentation. Denotenby "(ffink)*:_Csing,ffi*(k)(X_) ! 0(Z*
*)* its
composition with the inclusion Csing,ffi*(k)(X ) ! Csing*(X ). We obtain the fo*
*llowing
commutative diagram in chhfdOG (E).
lCsing,ffi1(k)*(__X)(A)linc//_lCsing,ffi2(k)*(__X)(A)linc//_lCsing,ffi3(k)*
**(__X)(A)linc//_. . .
|l"(ffi1k)*| l"(ffi2k)*|| |l"(ffi3k)*|
fflffl| fflffl| fflffl|
l0(Z)*(A)____id____//_l0(Z)*(A)__id____//_l0(Z)*(A)_id___//_
Here incdenotes the obvious inclusions. All arrows are_homotopy equivalences in
chhfdOG (E)nby_the_argument_above_since "*: Csing*(X ) ! 0(Z)* and each inclusi*
*on
Csing,ffi*(k)(X_) ! Csing*(X ) are chain homology equivalences by the contracti*
*bility
of X and Lemma 6.7 (ii). One easily checks that the upper row is an element in
cehhfdOG (E), namely echhfd(prk) O trans(A), and that the lower row is an eleme*
*nt
in echhfdOG (E), namely, the one given by A id-!A id-!A id-!. ...Hence we obtain
the desired natural transformation T whose evaluation at an object A is a weak
equivalence in echhfdOG (E).
7.Stability
In this section we will prove a stability result that implies (4.7).
Notation 7.1. Let E be a model for the classifying space EF G. Let (Xn, dn)n2N
be a sequence of quasi-metric spaces equipped with an isometric G-action. Denote
by ednthe product quasi-metric on Gx Xn defined by edn((g, x), (g0, x0)) = dG (*
*g, g0)+
dn(x, x0). We abbreviate
M
LG ((Xn, dn)n2N) = OG (E, Gx Xn, edn));
n2N
LG ((Xn, dn)n2N) = OG (E, (Gx Xn, edn)n2N).
The inclusion LG ((Xn, dn)n2N) ! LG ((Xn, dn)n2N) is a Karoubi filtration and we
denote the quotient by LG ((Xn, dn)n2N)> , its objects are the same as the obj*
*ects
26 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
of LG ((Xn, dn)n2N) but morphism are identified if they factor over an object in
LG ((Xn, dn)n2N).
Theorem 7.2. Let Xn, n 2 N be a sequence of simplicial complexes with a cell
preserving simplicial G-action. Suppose that there exists an N 2 N such that
dim Xn N for all n 2 N. For every n 2 N let dn be a quasi-metric on Xn
satisfying
dn(x, y) nd1(x, y) 8 x, y 2 Xn
with equality if x and y are contained in a common simplex of Xn. (Recall that *
*d1
denotes the l1-metric on simplicial complexes.) Assume that all isotropy groups*
* of
the action of G on Xn are contained in F. Then the inclusion
LG ((Xn, dn)n2N) ! LG ((Xn, dn)n2N)
induces an equivalence on the level of K-theory.
Note that (4.7)is a consequence of this Theorem. In this application to the
inclusion (3) in diagram (4.4)the quasi-metrics dn are equal to nd1, the l1-met*
*rics
scaled by n, but for the proof it will be convenient to also allow disjoint uni*
*ons of
simplicial complexes which carry a scaled l1-metric, but where different compon*
*ents
are infinitely far apart. We start by introducing some notation. Next we will s*
*tate
a special case and an excision result. The proof of Theorem 7.2 will then be an
easy induction.
Notation 7.3. Retain the assumptions of Theorem 7.2. Recall that dim Xn N.
Let Yn = q N be the disjoint union of the N-simplices of Xn. Equip Yn with
the quasi-metric d1n for which d1n(x, y) = nd1(x, y) if x and y are contained i*
*n a
common N-simplex and d1n(x, y) = 1 otherwise. Let @Yn = q@ N be the disjoint
union of the boundaries of the N-simplices of Yn and equip @Yn with the subspace
quasi-metric. Let X(N-1)nbe the (N -1)-skeleton of Xn equipped with the subspace
quasi-metric.
Proposition 7.4. Retain Notation 7.3. Then the K-theory of LG ((Yn, d1n)n2N)>
is trivial.
Proposition 7.5. Retain Notation 7.3. Then diagram induced from the pushout-
diagram that describes the attaching of the N-simplices
(7.6) LG ((@Yn, d1n)n2N)>______//_LG ((Yn, d1n)n2N)>
| |
| |
fflffl| fflffl|
LG ((X(N-1)n, dn)n2N)>____//LG ((Xn, dn)n2N)>
becomes homotopy cartesian after applying K-theory.
Proof of Theorem 7.2.Karoubi filtrations induce fibration sequences in K-theory,
[16]. Therefore
LG ((Xn, dn)n2N) ! LG ((Xn, dn)n2N) ! LG ((Xn, dn)n2N)>
induces a fibration sequence in K-theory. Hence the statement of the theorem
is equivalent to showing that the K-theory of LG ((Xn, dn)n2N)> vanishes. We
proceed by induction on N. If N = -1, then there is nothing to prove. Consider
(7.6). By the induction hypothesis the K-theory of the categories on the left b*
*oth
are trivial. By Proposition 7.4 the K-theory of the upper right category of (7.*
*6)
vanishes. Proposition 7.5 implies now that the K-theory of LG ((Xn, dn)n2N)>
has to vanish as well.
It remains to prove Propositions 7.4 and 7.5.
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 27
Proof of Proposition 7.4.This proof will be similar to the proof that homology
theories constructed using controlled algebra satisfy homotopy invariance and u*
*ses
an Eilenberg swindle.
We will construct for each n an Eilenberg-swindle on OG (E, Gx Yn, ed1n). Si*
*nce
the construction leaves the G x Yn-direction untouched it will be clear that th*
*ese
Eilenberg-swindles can be combined to an Eilenberg-swindle on LG ((Yn, d1n)n2N)*
*> .
If E = pt, then we can define this swindle by pushing along the [1, 1)-directio*
*n,
compare Lemma 3.6 (ii). In the general case, we will also need to use contracti*
*ons
in E towards fixed points for isotropy groups.
Fix n 2 N.
Let Rn be the set of N-simplices of Yn. The isotropy groups of Rn agree with*
* the
isotropy groups of Yn and are thus all contained in F. By the universal property
of E there exists a G-equivariant map ': Rn ! E and a G-equivariant homotopy
h: Rnx E x [0, 1] ! E with h0(r, e) = e and h1(r, e) = '(r). Denote by p: Yn ! *
*Rn
the canonical projection map that collapses each N-simplex to a point. For k 2 *
*N0
the map
(g, y, e, t) 7! (g, y, hk=(k+t)(p(y), e), t + k)
where g 2 G, y 2 Yn, e 2 E, t 2 [1,L1) induces a functor Sk from OG (E, Gx Yn, *
*ed1n)
to itself. Our first claim is that 1k=0Sk also yields a well defined functor.*
* There is
a canonical natural transformation ok between Sk and Sk+1,Lsee (3.1).LOur second
claim is that 1k=0ok yields a natural equivalence from 1k=0Sk to 1k=1Sk. S*
*ince
S0 = idthis gives the desired Eilenberg-swindle.
It remains to prove the two claims above. In both cases the only nontrivial *
*claim
is that the continuous control condition (3.2)is preserved.
We start withLthe first claim. Let ' be a morphism in OG (E, Gx Yn, ed1n). T*
*he
support of ( 1k=0Sk)(') is given by all pairs of points in Gx Ynx Ex [1, 1) of*
* the
form
((g, y, hk=(k+t)(p(y), e), t + k), (g0, y0, hk=(k+t)(p(y0), e0), t0+ k*
*)),
where k 2 N0 and ((g, y, e, t), (g0, y0, e0, t0)) 2 supp '. Let U be an G~e-in*
*variant
open neighborhood of ~e2 E and ~ > 0. We need to show that there is an G~e-
invariant neighborhood V of ~eand oe > ~ such that if ((g, y, e, t), (g0, y0, e*
*0, t0)) 2
supp ', k 2 N0, hk=(k+t)(p(y), e) 2 V and t > oe, then hk=(k+t0)(p(y0), e0) 2 U*
* and
t0 > ~. By [3, Proposition 3.4] there is aTsequence_V 1 V 2 . . .of open G~e-
invariant neighborhoods of ~e2 E such that 1l=1GV l= G~eand gV l\ V l= ; if
g 2 G - G~e. We proceed now by contradiction and assume that for every l there *
*is
Pl= ((gl, yl, el, tl), (g0l, y0l, e0l, t0l)) 2 supp' and kl2 N0 such that
(hkl=(kl+tl)(p(yl), el), tl+2kl)V lx(~ + l, 1)
but (hkl=(kl+t0l)(p(y0l), e0l),6t0l+2kl)Ux (~, 1).
The metric control with respect to ed1nfor the morphism ' implies that p(yl) = *
*p(y0l)
for all sufficiently large l. From the metric control condition with respect t*
*o the
projection to [1, 1), see (3.3), we conclude that |tl- t0l| ff for some ff > 0
independent of l. Since tl+ kl> ~ + l this implies t0l+ kl> ~ for sufficiently *
*large
l. Therefore we may assume that
(7.7) hkl=(kl+t0l)(p(y0l), e0l) 62 U 8 l.
Passing to a subsequence, if necessary, we can assume that kl=(kl+tl) and kl=(k*
*l+t0l)
both converge for l ! 1. Since tl+ kl> ~ + l we conclude from
fifi fi fi 0 fi
fifikl_- __kl__fifi= fifikl._(tl-_tl)_fifi __ff__ _ff__
kl+ tl kl+ t0lfi fi(kl+ tl) . (kl+ft0l)i kl+ tl ~ + l
28 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
that both sequences have the same limit s 2 [0, 1]. Because the morphism ' has
G-compact support with respect to the projection to Gx Ynx E, we can assume
that there are al2 G and ("g, "y, "e) such that
al(gl, yl, el) ! ("g, "y, "e) as l ! 1.
Now alhkl=(kl+tl)(p(yl), el) = hkl=(kl+tl)(p(alyl), alel) converges to hs(p("y)*
*, "e). Since
hkl=(kl+tl)(p(yl), el) 2 V lwe conclude hs(p("y), "e) = a"efor some a 2 G and a*
*-1al2
G~efor sufficiently large l from the properties of the V l. Because U and the V*
* lare
G~einvariant and supp' is G-invariant, we can replace Pl by a-1alPl. Therefore
we may now assume that
(gl, yl, el) ! ("g, "y, "e) as l ! 1.
Because Rl is discrete we can also assume that p(yl) = p(y0l) = p("y) for all l*
*. Let
U" E be the preimage of U E under the G-equivariant map e 7! hs(p(y), e).
Now we use that supp' satisfies the continuous control condition (3.2)to conclu*
*de
that there exists an open Ge-invariant neighborhood W of e 2 E and oe > 0
such that if ((g, y, e, t), (g0, y0, e0, t0)) 2 supp ', e 2 W , t > oe and t0 >*
* ~ then
e02 "U. Since el ! "ewe can apply this to Pl 2 supp' and conclude that e0l2 "U
for sufficiently large l. Thus hs(p(y), e0l) 2 U for sufficiently large l. Bu*
*t this
contradicts (7.7)since kl=(kl+ tl) ! s as l ! 1 and p(y) = p(y0l) for all l. Th*
*is
finishes the proof of the first claim.
For the second claim will use a similar argument. Let A be an object of
OG (E, Gx Yn, ed1n). Then the support of the isomorphism
M1 M1 1M
( ok)(A): Sk(A) ! Sk(A)
k=0 k=0 k=1
is the set of all pairs of points in Gx Ynx Ex [1, 1) of the form
((g, y, hk=(k+t)(p(y), e), t + k), (g, y, hk+1=(k+1+t)(p(y), e), t + k +*
* 1))
where k 2 N0 and (g, y, e, t) 2 suppA, compare (3.1). We need to show that this
set satisfies the continuous control condition (3.2). Let U be an G~e-invariant*
* open
neighborhood of ~e2 E and ~ > 0. Let V lbe a sequence of open neighborhoods
as used in the proof of the first claim. We proceed as before by contradiction *
*and
assume that for every l there is (gl, yl, el, tl) 2 suppA and kl2 N0 such that
(hkl=(kl+tl)(p(yl), el), tl+ kl) 2 V lx(~ + l, 1)
but (hkl+1=(kl+1+tl)(p(yl), el), tl+ kl+61)2Ux (~, 1).
(Strictly speaking we also need to consider the case where we interchange kland*
* kl+
1 because of (3.4), but this case can be treated by essentially the same argume*
*nt.)
From tl+ kl+ 1 > ~ we conclude
(7.8) hkl+1=(kl+1+tl)(p(yl), el) 62 U 8 l.
Passing to a subsequence, if necessary, we can assume that kl=(kl+ tl) and kl+
1=(kl+ 1 + tl) both converge. As above we conclude that both sequences have the
same limit s 2 [0, 1]. Because the object A has G-compact support with respect *
*to
the projection to Gx Ynx E, we can assume that there are al2 G such that
al(gl, yl, el) ! ("g, "y, "e) as l ! 1.
By a similar argument as above we can in fact assume that al is trivial. Since
~e= liml!1hkl=(kl+tl)(p(yl), el) = liml!1hkl+1=(kl+1+tl)(p(yl), el)
we obtain a contradiction to (7.8).
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 29
Before we can prove Proposition 7.5 we will need to unravel the definition o*
*f the
categories appearing in (7.6)and introduce some more notation.
Notation 7.9. Retain Notation 7.3. Let
a a
B := Gx @Ynx Ex [1, 1), Y := Gx Ynx Ex [1, 1),
n2Na n2Na
A := Gx X(N-1)nxEx [1, 1), X := Gx Xnx Ex [1, 1).
n2N n2N
` ~n
Let FB be the collection of subsets of B of the form n=1Gx @Ynx Ex [1, 1) for
some ~n2 N. Similar we have collections FA , FY and FX respectively of subsets
of A,`Y and X respectively. Let FBcsbe the collection of subsets of B of the
form n2NKn, where each Kn is the preimage of a G-compact subset under the
projection Gx @Ynx Ex [1, 1) ! Gx @Ynx E. Similar we have collections FAcs, FYcs
and FXcsrespectively of subsets of A, Y and X respectively. Let EB be the colle*
*ction
of subsets J Bx B satisfying the following conditions:
`
(7.10) J n2NJn with respect to the canonical inclusion
a
(Gx @Ynx Ex [1, 1))x2 ! Bx B,
n2N
where for every n 2 N the set Jn (Gx @Ynx Ex [1, 1))x2 is such that Jn
is the preimage of some J0n2 EEGccwith respect to the canonical project*
*ion
Gx @Ynx Ex [1, 1) ! Ex [1, 1), compare Section 3.2;
(7.11) There is ff > 0 such that ((g, y, e, t), (g0, y0, e0, t0)) 2 J with g, *
*g02 G, y, y02
@Yn, e, e02 E and t, t02 [0, 1) implies d1n(y, y0) ff and dG (g, g0) *
* ff.
Similar we have collections EA , EY and EX respectively of subsets of Ax2, Y x2*
*and
Xx 2respectively. Of course we use the quasi-metrics dn in the definition of EA*
* and
EX .
With this notation diagram (7.6)becomes
(7.12) CG (B, EB , FBcs; A)>FB__//_CG (Y, EY , FYcs; A)>FY
| |
| |
fflffl| A fflffl| X
CG (A, EA , FAcs; A)>F__//CG (X, EX , FXcs; A)>F
where we used a more general germ notation to denote Karoubi quotients, see for
instance [8, Section 2.1.6]. For example, for the upper left category this just*
* means
that morphisms are identified if their difference factors over an object whose *
*support
lies in some F 2 FB . We will drop A from the notation.
For ff > 0 and F 2 FAcslet F ffbe the subset of X consisting of all points
(g, x, e, t) 2 X with the property that if x 2 Xn then there is x0 2 X(N-1)nwith
(g, x0, e, t) 2 F and dn(x, x0) ff. Define FXA as the collection of all subse*
*ts of X
of the form F fffor all ff > 0, F 2 FAcsand F 02 FXcs. We will follow [8, Secti*
*on 8.4]
and abuse notation to denote by FXcs\ FXA the collection of all subsets of the *
*form
F \ F 0with F 2 FXcsF 02 FXA Similar definitions yield FYB, a collection of sub*
*sets
of Y and FYcs\ FYB.
Lemma 7.13. Retain Notation 7.9. The inclusions
B G Y Y Y >FY
CG (B, EB , FBcs)>F! C (Y, E , Fcs\ FB ) ,
A G X X X >FX
CG (A, EA , FAcs)>F! C (X, E , Fcs\ FA )
are equivalences of categories.
30 ARTHUR BARTELS, WOLFGANG L"UCK, AND HOLGER REICH
Proof. It is a formal consequence of the definitions that both functors yield i*
*somor-
phisms on morphism groups. It remains to show that every object in the target
category is isomorphic to an object in the image of the functor.XWe consider the
second functor. Let M be an object in CG (X, EX , FXcs\ FXA)>F . By definition
supp M is a locally finite subset of F ff\ F 0for some ff > 0, F 2 FAcs, F 02 F*
*Xcs.
Therefore there is a G-equivariant map f :supp M ! F with the property that if
f(g, x, e, t) = (g0, x0, e0, t0) with x 2 Xn then g0 = g, e0 = e, t0 = t, x0 2 *
*Xn and
dn(x, x0) ff. (The map is not canonical; we have to choose x0for every x in a*
* G-
equivariant way.) It is not hard to see that f is finite-to-one and has a local*
*ly finite
*
* A
image. Thus we can apply f to M to obtain an object f*(M) of CG (A, EA , FAcs)>*
*F .
Clearly, {(s, f(s)) | s 2 suppM} 2 EX . Thus M and f*(M) are isomorphic.
The first functor can be treated similarly.
Proof of Proposition 7.5.Retain Notation 7.9. Because of (7.12)and Lemma 7.13
it suffices to prove that
__Y__//_G Y Y >FY
CG (Y, EY , FYcs\ FYB)>F C (Y, E , Fcs)
| |
| |
fflffl| X fflffl| X
CG (X, EX , FXcs\ FXA)>F____//CG (X, EX , FXcs)>F
yields a homotopy cartesian diagram in K-theory.
The two rows of this diagram are now Karoubi filtrations and on the quotients
we obtain an induced functor
Y [FY G X X >FX [FX
(7.14) CG (Y, EY , FYcs)>F B ! C (X, E , Fcs) A .
(Here we are again abusing notation following [8, Section 8.4]: FY [ FYB is the
collection of all sets of the form F [ F 0with F 2 FY and F 02 FYB and the
definition of FX [ FXA is similar.)
Because Karoubi filtrations induce fibration sequences in K-theory [16], it *
*suffices
to show that (7.14)is an equivalence of categories. Because the canonical map
Yn ! Xn induces a homeomorphism Yn - @Yn ! Xn - X(N-1)nevery object in the
target category is isomorphic to an object in the image. Hence it suffices to s*
*how
that (7.14)is full and faithful.
Every morphism in the category CG (X, EX , FXcs; A) can be written`as the su*
*m of
two morphisms '+_, where ' does not connect different k-simplices of n2NX(k)n
and _ has no component that connects two points on the same simplex. Clearly, '
can be lifted to CG (Y, EY , FYcs). It follows from Lemma 7.15 below that _ can*
* be
factored over an object whose support is contained in some F 2 FXA. The definit*
*ion
X [FX
of the Karoubi quotient implies that _ is trivial in CG (X, EX , FXcs)>F A .*
* There-
fore (7.14)is surjective on morphism sets. The injectivity on morphism sets fol*
*lows
from the fact that the preimage of an F 2 FXA is contained in some F 02 FYB.
Lemma 7.15. Let Z be an n-dimensional simplicial complex. If is an n-simplex
in Z, x 2 , y 2 Z - then there is z 2 @ such that d1(x, z) 2d1(x, y). (He*
*re
d1 denotes the l1-metric on Z).
Proof.Let 0 be the simplex uniquely determined by the property that y lies in
its interior. Then \ 06= 0. Let xi, i 2 I be the barycentric coordinates of*
* x
and yi0, i02 I0 be the barycentric coordinates of y, where I and I0 respectivel*
*y are
the vertices of and 0. We can assume x =2@ , because otherwise we simply set
z = x. Therefore xi 6= 0 for all i 2 I. Since \ 0 6= 0 there exists an i0 2*
* I
with i0 =2I0. We have xi06= 0 and xi0 d1(x, y). Now let z 2 @ be the point
THE K-THEORETIC FARRELL-JONES CONJECTURE FOR HYPERBOLIC GROUPS 31
with coordinates zi = _xi__1-xi0if i 6= i0 and zi0= 0. Then d1(x, z) = 2xi0and *
*hence
d1(x, z) 2d1(x, y).
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Westf"alische Wilhelms-Universit"at M"unster, Mathematisches Institut, Einst*
*einstr. 62,
D-48149 M"unster, Germany
E-mail address: bartelsa@math.uni-muenster.de
URL: http://www.math.uni-muenster.de/u/bartelsa
E-mail address: lueck@math.uni-muenster.de
URL: http://www.math.uni-muenster.de/u/lueck
E-mail address: reichh@math.uni-muenster.de
URL: http://www.math.uni-muenster.de/u/reichh