Amenable actions and applications

18 Pages·2006·0.14 MB·English

Amenable actions and applications
NarutakaOzawa
Abstract. We will give a brief account of (topological) amenable actions and exactness for
countable discrete groups. The class of exact groups contains most of the familiar groups
and yet is manageable enough to provide interesting applications in geometric topology, von
Neumannalgebrasandergodictheory.
MathematicsSubjectClassification(2000). Primary46L35;Secondary20F65,37A20,43A07.
Keywords.Amenableactions,exactgroups,groupvonNeumannalgebras.
1. Introduction
The notion of amenable groups was introduced by J. von Neumann in 1929 in his
investigationoftheBanach–Tarskiparadox. Heobservedthatnon-abelianfreegroups
arenotamenableandthatthisfactisthesourceoftheBanach–Tarskiparadox. Since
thenithasbeenshownthattheamenabilityofalocallycompactgroupisequivalentto
manyfundamentalpropertiesinharmonicanalysisofthegroup: theFølnerproperty,
thefixedpointpropertyandtheweakcontainmentofthetrivialrepresentationinthe
regularrepresentation,tonameafew. Foradiscretegroup,amenabilityofthegroupis
∗
alsocharacterizedbynuclearityofitsgroupC -algebra,andbyinjectivityofitsgroup
von Neumann algebra. In this note, we are mainly interested in countable discrete
groups. Theclassofamenablegroupscontainsallsolvablegroupsandisclosedunder
subgroups, quotients, extensions and directed unions. As we mentioned before, a
non-abelianfreegroup, oranygroupwhichcontainsit, isnotamenable. Amenable
groupsplayapivotalroleinthetheoryofoperatoralgebras. Manysignificantoperator
algebra-relatedproblemsongroupshavebeensolvedforamenablegroups. Wejust
citetwoofthem;theclassificationofgroupvonNeumannalgebras[14]andmeasure
equivalences[16],[58]ontheonehand,andtheBaum–Connesconjecture[46]onthe
otherhand. Inrecentyears,therehavebeenexcitingbreakthroughsinbothsubjects
beyondtheamenablecases. Wereferto[68]and[84]foraccountsofthisprogress.
We will also treat the classification of group von Neumann algebras and measure
equivalences in Section 4. Since many significant problems, if not all, are already
solvedforamenablegroups,wewouldliketosetoutfortheworldofnon-amenable
groups. Still,asGromov’sprinciplegoes,nostatementaboutallgroupsisbothnon-
trivialandtrue. Sowewantagoodclassofgroupstoplaywith. Weconsideraclass
ProceedingsoftheInternationalCongress
ofMathematicians,Madrid,Spain,2006
©2006EuropeanMathematicalSociety
1564 NarutakaOzawa
asgoodifitcontainsmanyofthefamiliarexamples,ismanageableenoughsothatit
maintainsnon-trivialtheorems,andcanbecharacterizedinvariouswayssothatitis
versatile. Webelievethattheclassofexactgroups, whichwillbeintroducedinthe
∗
followingsection,standsthesetests. ThestudyofexactnessoriginatesinC -algebra
theory [50], [51], [52] and was propagated to groups. The class of exact groups is
fairly large and it contains all amenable groups, linear groups [39] and hyperbolic
groups[2],tonameafew. Itisclosedundersubgroups,extensions,directedunions
and amalgamated free products. (Since every free group is exact and there exists a
non-exactgroup[37],aquotientofexactgroupneedsnotbeexactunlessthenormal
subgroupisamenable.) Moreover,thereisaremarkabletheoremthattheinjectivity
part of the Baum–Connes conjecture holds for exact groups [45], [76], [83], [84].
SincethispartoftheBaum–Connesconjecturehasalotofapplicationsingeometry
andtopology,includingthestrongNovikovconjecture,itisaninterestingchallenge
to prove exactness of a given group. We will encounter some other applications in
vonNeumannalgebratheoryandergodictheoryinSection4.
2. Amenableactionsandexactness
We first review the definition of and basic facts on amenable actions. We refer to
[65] for the theory of amenable groups and to [5], [11] for the theory of amenable
actions. The notion of amenability for a group action was first introduced in the
measure space setting in the seminal paper [85], which has had a great influence in
both ergodic theory and von Neumann algebra theory. In this spirit the study of its
topologicalcounterpartwasinitiatedin[3]. Inthisnote, werestrictourattentionto
continuousactionsofcountablediscretegroupson(notnecessarilysecondcountable)
compactspaces. AlltopologicalspacesareassumedtobeHausdorffandallgroups,
writtenas(cid:2),(cid:3),...,areassumedtobecountableanddiscrete. Let(cid:2) beagroup. A
(topological)(cid:2)-spaceisatopologicalspaceXtogetherwithacontinuousactionof(cid:2)
onit;(cid:2)×X (cid:3)(s,x)(cid:4)→s.x ∈X. Foragroup(oranycountableset)(cid:2),welet
(cid:2) (cid:3) (cid:4)
prob((cid:2))= μ∈(cid:4) ((cid:2)):μ≥0, μ(t)=1 ⊂(cid:4) ((cid:2))
1 t∈(cid:2) 1
andequipprob((cid:2))withthepointwiseconvergencetopology. Wenotethatthistopol-
ogy coincides with the norm topology. The space prob((cid:2)) is a (cid:2)-space with the
(cid:2)-actiongivenbythelefttranslation: (s.μ)(t)=μ(s−1t).
Definition2.1. Wesaythatacompact(cid:2)-spaceX isamenable(or(cid:2) actsamenably
onX)ifthereexistsasequenceofcontinuousmaps
μ : X (cid:3)x (cid:4)→μx ∈prob((cid:2))
n n
suchthatforeverys ∈(cid:2) wehave
lim sup(cid:10)s.μx −μs.x(cid:10)=0.
n→∞x∈X n n
Amenableactionsandapplications 1565
When X is a point, the above definition degenerates to one of the equivalent
definitions of amenability for the group (cid:2). Moreover, if (cid:2) is amenable, then every
(cid:2)-spaceisamenable. Conversely,ifthereexistsanamenable(cid:2)-spacewhichcarries
aninvariantRadonprobabilitymeasure,then(cid:2)itselfisamenable. IfXisanamenable
(cid:2)-space, thenX isamenableasa(cid:3)-spaceforeverysubgroup(cid:3). Itfollowsthatall
isotropysubgroupsofanamenable(cid:2)-spacehavetobeamenable. Werecallthatthe
isotropy subgroup of x in a (cid:2)-space X is {s ∈ (cid:2) : s.x = x}. It is also easy to see
that if there exists a (cid:2)-equivariant continuous map from a (cid:2)-space Y into another
(cid:2)-space X and if X is amenable, then so is Y. Finally, we only note that there are
severalequivalentcharacterizationsofanamenableactionwhichgeneralizethosefor
anamenablegroup.
Manyamenableactionsnaturallyarisefromthegeometryofgroups. Thefollow-
ingarethemostbasicexamplesofamenableactions.
Example 2.2. Let F = (cid:11)g ,...,g (cid:12) be the free group of rank r < ∞. Then its
r 1 r
(Gromov)boundary∂F isamenable. WenotethattheCayleygraphofF w.r.t.the
r r
standardsetofgeneratorsisasimplicialtreeanditsboundary
∂F ⊂{g ,g−1,...,g ,g−1}N
r 1 1 r r
isdefinedasthecompacttopologicalspaceofallinfinitereducedwords,equippedwith
therelativeproducttopology(seeFigure1). Similarly,withanappropriatetopology,
F ∪ ∂F becomes a compactification of F . The free group F acts continuously
r r r r
on∂F byleftmultiplication(andrectifyingpossibleredundancy). Forx ∈∂F with
r r
itsreducedformx = a a ...,wesetx = e andx = a ...a . Foreveryn ∈ N,
1 2 0 k 1 k
welet
(cid:5)n−1
1
μ : ∂F (cid:3)x (cid:4)→μx = δ ∈prob(F ).
n r n n xk r
k=0
Thusμx isthenormalizedcharacteristicfunctionofthefirstnsegmentsofthepathin
n
theCayleygraphofF ,connectingetox (seeFigure1). Itisnothardtoseethatμ
r n
isacontinuousmapsuchthat
2|s|
sup (cid:10)s.μx −μs.x(cid:10)≤
x∈∂Fr n n n
for every s ∈ F , where |s| is the word length of s. Indeed, s.μx is the normalized
r n
characteristicfunctionofthefirstnsegmentsofthepathconnectings tos.x,which
hasalargeintersectionwiththepathconnectingetos.x (seeFigure2).
There are generalizations of this construction to groups acting on more general
buildings[72]andonhyperbolicspaces[2].
Example2.3. Let(cid:2)beadiscretesubgroupofthespeciallineargroupSL(n,R)(e.g.,
(cid:2) =SL(n,Z))andP ⊂SL(n,R)betheclosedsubgroupofuppertriangularmatrices.
1566 NarutakaOzawa
(cid:3)s.x
(cid:2) (cid:3)x ∈∂F (cid:6)
(cid:2) (cid:2) (cid:2) (cid:2) 2 (cid:3)
(cid:5)
x (cid:3)
(cid:2) 2(cid:2) (cid:5)
(cid:2) (cid:2) (cid:2)e x1(cid:2) (cid:2) s.μx (cid:5) μsn.x
(cid:2) (cid:2) n (cid:5)
(cid:3)(cid:3) (cid:3)(cid:3)(cid:4)
s (cid:4)(cid:3)
(cid:2) (cid:2) (cid:2)
e
(cid:2) F
2
Figure 1. The Cayley graph of F2 and the Figure2.Amenabilityof∂F2.
boundary∂F .
2
Thentheleftmultiplicationactionof(cid:2) ontheFurstenbergboundarySL(n,R)/P is
amenable. More generally, if G is a locally compact group with a closed amenable
locallycompactsubgroupP (suchthatG/P compact),theneverydiscretesubgroup(cid:2)
ofGactsamenablyonG/P.
Afar-reachinggeneralizationofthisexampleisgivenin[39], whereitisshown
thatanylineargroupadmitsanamenableactiononsomecompactspace. Thusmany
non-amenablegroupsadmitamenableactions.
Definition2.4. Wesayagroup(cid:2)isexactifthereexistsacompact(cid:2)-spaceXwhich
isamenable.
Exact groups are also said to be boundary amenable, amenable at infinity or to
have the property A. By definition, all amenable groups are exact. Let X be a
compact(cid:2)-space. Then,bytheuniversalityoftheStone–Cˇechcompactificationβ(cid:2),
thereexistsa(cid:2)-equivariantcontinuousmapfromβ(cid:2)intoX. Itfollowsthat(cid:2)isexact
iffβ(cid:2) (ortheboundary∂β(cid:2) = β(cid:2)\(cid:2))isamenable. Moreover,whether(cid:2) isexact
or not, β(cid:2) is amenable as a (cid:3)-space for every exact subgroup (cid:3) of (cid:2) since there
exists a (cid:3)-equivariant continuous map from β(cid:2) into β(cid:3). This observation implies
thatexactnessispreservedunderadirectedunion,i.e.,agroup(cid:2)isexactiffallofits
finitelygeneratedsubgroupsareexact.
AmenabilityoftheStone–Cˇechcompactificationβ(cid:2) leadstoanintrinsiccharac-
terization of an exact group (cid:2). Before stating it, we introduce the notion of coarse
metricspaces[36]. Letd bealefttranslationinvariantmetricon(cid:2) whichisproper
in the sense that every subset of finite diameter is finite. Then l(s) = d(s,e) is a
lengthfunctionon(cid:2), i.e., l(s−1) = l(s), l(st) ≤ l(s)+l(t)foreverys,t ∈ (cid:2), and
l(s)=0iffs =e. Thelengthfunctionlisproperinthesensethatl−1([0,R])isfinite
for every R > 0. Conversely, every proper length function l gives rise to a proper
left translation invariant metric d on (cid:2) such that d(s,t) = l(s−1t). If S is a finite
Amenableactionsandapplications 1567
generatingsubsetof(cid:2),thenthecorrespondingwordmetricisdefinedby
dS(s,t)=min{n:s−1t =s1...sn, si ∈S∪S−1}.
Wenotethatevenwhen(cid:2)isnotfinitelygenerated,thereexistsaproperlefttranslation
invariant metric d on (cid:2) (as we assume that (cid:2) is countable). Two proper length
functionslandl(cid:15)areequivalentinthesensethatl(s )→∞iffl(cid:15)(s )→∞. Thuswe
n n
areleadtothenotionofcoarseequivalence,whichisaveryloosenotion. Twometric
(cid:15) (cid:15)
spaces (X,d) and (X,d ) are coarsely isomorphic if there exists a (not necessarily
continuous)mapf: X →X(cid:15) suchthatd(z,f(X))<∞foreveryz ∈X(cid:15) and
ρ−(d(x,y))≤d(cid:15)(f(x),f(y))≤ρ+(d(x,y))
for some fixed function ρ± on [0,∞) with limr→∞ρ−(r) = ∞. Such f is called
a coarse isomorphism. We observe that any two proper left translation invariant
(cid:15)
metrics d and d on (cid:2) are coarsely equivalent in the sense that the formal identity
(cid:15)
map from ((cid:2),d) onto ((cid:2),d ) is a coarse isomorphism. A coarse metric space is a
spacetogetherwithacoarseequivalenceclassofmetrics. Hence,(cid:2)isprovidedwith
(cid:15)
auniquecoarsemetricspacestructure. Twogroups(cid:2) and(cid:2) aresaidtobecoarsely
isomorphic iftheyarecoarselyisomorphicascoarsemetricspaces. Itfollowsfrom
thefollowingtheoremthatexactnessisacoarseisomorphisminvariant. Inparticular,
agroupisexactifithasafiniteindexsubgroupwhichisexact.
Theorem2.5([47],[83]). Foragroup(cid:2),thefollowingareequivalent.
1. Thegroup(cid:2) isexact.
2. Themetricspace((cid:2),d)hasthepropertyA: Foreveryε >0andR >0,there
exist a map ν: (cid:2) → prob((cid:2)) and S > 0 such that (cid:10)ν −ν (cid:10) ≤ ε for every
s t
s,t ∈(cid:2) withd(s,t)<R andsuppν ⊂{t :d(s,t)<S}foreverys ∈(cid:2).
s
3. Foreveryε > 0andR > 0,thereexistaHilbertspaceH,amapξ: (cid:2) → H
andS >0suchthat|1−(cid:11)ξ ,ξ (cid:12)|<εforeverys,t ∈(cid:2) withd(s,t)<R and
t s
(cid:11)ξ ,ξ (cid:12)=0foreverys,t ∈(cid:2) withd(s,t)≥S.
t s
Moreover,if (cid:2)isexact,then(cid:2)iscoarselyisomorphictoasubsetofaHilbertspace.
Themainresultof[83]istheinjectivitypartoftheBaum–Connesconjecturefor
a group which is coarsely embeddable into a Hilbert space. (See also [45], [76],
[84].) This justifies the study of exactness for groups. It is not known whether or
not coarse embeddability into a Hilbert space implies exactness (even in the case
of groups with the Haagerup property). We recall that a metric space (X,d) has
asymptotic dimension ≤ d [36] if for every R > 0, there exists a covering U of X
such that supU∈Udiam(U) < ∞ and |{U ∈ U : U ∩B (cid:17)= ∅}| ≤ d +1 for any
subset B ⊂ X with diam(B) < R. Asymptotic dimension is a coarse equivalence
invariantandhenceaninvariantforagroup. WenotethatthegroupsZd andFd have
r
asymptoticdimensiond.
1568 NarutakaOzawa
Corollary2.6([47]). Acoarsemetricspacewithfiniteasymptoticdimensionhasthe
propertyA. Inparticular,agroupwithfiniteasymptoticdimensionisexact.
Itwasshownin[22]thateveryCoxetergrouphasfiniteasymptoticdimensionand
henceisexact. Wereferto[8]formoreinformationonasymptoticdimension.
Wedescribearelativeversionofanamenableaction,whichisusefulinproving
variouskindsofpermanencepropertiesofexactness. Thereareotherapproaches[6],
[7],[20]whichareaswelluseful. Thefollowingisinthespiritof[3].
Proposition2.7([63]). LetXbeacompact(cid:2)-spaceandK beacountable(cid:2)-space.
AssumethatthereexistsanetofBorelmaps
μ : X →prob(K)
n
(i.e.,thefunctionX (cid:3)x (cid:4)→μx(a)∈RisBorelforeverya ∈K)suchthat
n
(cid:6)
lim (cid:10)s.μx −μs.x(cid:10)dm(x)=0
n n
n
X
for every s ∈ (cid:2) and every Radon probability measure m on X. Then (cid:2) is exact
provided that all isotropy subgroups of K are exact. Indeed, if Y is a compact
(cid:2)-spacewhichisamenableasa(cid:3)-spaceforeveryisotropysubgroup(cid:3),thenX×Y
(withthediagonal(cid:2)-action)isanamenable(cid:2)-space.
Corollary2.8([52]). Anextensionofexactgroupsisagainexact.
Proof. If(cid:3)(cid:2)(cid:2)isanormalsubgroupsuchthat(cid:2)/(cid:3)isexact,thenProposition2.7is
applicabletoanamenablecompact((cid:2)/(cid:3))-spaceXandK =(cid:2)/(cid:3) (cid:3)
We turn our attention to a group acting on a countable simplicial tree T, which
may not be locally finite. We will define a compactification T = T ∪∂T of T, to
which Proposition 2.7 is applicable. We recall that a simplicial tree is a connected
graphwithoutnon-trivialcircuits,andidentifyT withitsvertexset. Theboundary∂T
of T is defined as in Example 2.2. Thus ∂T is the set of all equivalence classes of
(one-sided)infinitesimplepathsinT,wheretwoinfinitesimplepathsareequivalentif
theirintersectionisinfinite. Foreverya ∈T andx ∈∂T,thereexistsauniqueinfinite
simple path γ in the equivalence class x which starts at a. We say that the path γ
connectsatox. ItfollowsthateverytwodistinctpointsinT =T ∪∂T areconnected
byauniquesimplepath(whichisabiinfinitepath,withtheobviousdefinition,when
bothpointsareboundarypoints). EveryedgeseparatesT intotwocomponents,and
every finite subset of edges separates X into finitely many components. Now we
equip T with a topology by declaring that all such components are open. It turns
out that T is compact with this topology. We note that T is dense but not open
in T (unless T is locally finite) and that every automorphism s of T extends to a
Amenableactionsandapplications 1569
homeomorphism on T. Fixing a base point e ∈ T, we define μ : ∂T → prob(T)
n
exactlyasinExample2.2. Itisnothardtoseethatμ isaBorelmapsuchthat
n
2d(s.e,e)
sup (cid:10)s.μx −μs.x(cid:10)≤
x∈∂T n n n
foreveryautomorphisms ofT (cf.Figure2). Weextendμ toT bysimplyletting
n
μa =δ ∈prob(T)fora ∈T. ThenthesequenceofBorelmapsμ : T →prob(T)
n a n
satisfiestheassumptionofProposition2.7forX =T,K =T andanygroup(cid:2)acting
onT.
We recall that associated with the fundamental group of a graph of groups there
exists a tree, called the Bass–Serre tree, on which the group acts. We describe it in
the case of an amalgamated free product. Let (cid:2) = (cid:2) ∗ (cid:2) be the amalgamated
1 (cid:3) 2
freeproductofgroups(cid:2) and(cid:2) withacommonsubgroup(cid:3). Thentheassociated
1 2
Bass–Serre tree T is the disjoint union (cid:2)/(cid:2) (cid:19)(cid:2)/(cid:2) of left cosets, where s(cid:2) and
1 2 1
t(cid:2) areadjacentifs(cid:2) ∩t(cid:2) (cid:17)= ∅. ThustheedgesetofT coincideswith(cid:2)/(cid:3),and
2 1 2
anedges(cid:3)connectss(cid:2) ands(cid:2) . ItturnsoutthatT isatree. Thegroup(cid:2)actsonT
1 2
fromtheleftinsuchawaythateachvertexstabilizerisconjugatetoeither(cid:2) or(cid:2)
1 2
andeachedgestabilizerisconjugateto(cid:3). WenotethatthetreeT isnotlocallyfinite
unless(cid:3)hasfiniteindexinboth(cid:2) and(cid:2) .
1 2
Corollary2.9([25],[78]). Let(cid:2)beagroupactingonacountablesimplicialtreeT.
Then (cid:2) is exact provided that all isotropy subgroups are exact. In particular, an
amalgamatedfreeproductandanHNN-extensionofexactgroupsareagainexact.
It follows that one-relator groups are exact [38] because they are made up by
using HNN-extensions following the McCool–Schupp algorithm. A similar remark
applies to a fundamental group of a Haken 3-manifold thanks to the Waldhausen
decomposition.
Example2.2canbegeneralizedtoahyperbolicspace,too. Thenotionofhyper-
bolicity was introduced in the very influential paper [35] and has been extensively
studied since. A metric space is said to be hyperbolic if it is “tree-like” in certain
sense, and a finitely generated group (cid:2) is said to be hyperbolic if its Cayley graph
ishyperbolic. Hyperbolicityisarobustnotionandtherearemanynaturalexamples
of hyperbolic groups including the free groups. Every hyperbolic group has a nice
compactification, called the Gromov compactification, which is a generalization of
that given in Example 2.2. It is shown in [2] that the action of a hyperbolic group
onitsGromovcompactificationisamenable. (Seealso[9]andtheappendixof[5].)
Theresultisgeneralizedin[48],[63]toagroupactingonhyperbolicspaces,which
arenotnecessarilylocallyfinite. Compactificationofanon-locally-finitehyperbolic
graphwasconsideredin[10], whereitsBowditchcompactificationK isintroduced
forafinehyperbolicgraphK. AsimplicialtreeT anditscompactificationT arethe
simplestnon-trivialexamplesofauniformlyfinehyperbolicgraphanditsBowditch
1570 NarutakaOzawa
compactification. See [10] for details. As in the case for a simplicial tree, the as-
sumptionofProposition2.7issatisfiedforauniformlyfinehyperbolicgraphK,its
BowditchcompactificationK andanygroupactingonK [63]. Byacharacterization
ofarelativelyhyperbolicgroup[10],weobtainthefollowingcorollary.
Corollary2.10([20],[59],[63]). Arelativelyhyperbolicgroupisexactprovidedthat
allperipheralsubgroupsareexact. Inparticular,everyhyperbolicgroupisexact.
Examplesofrelativelyhyperbolicgroupsincludethefundamentalgroupsofcom-
plete non-compact finite-volume Riemannian manifolds with pinched negative sec-
tionalcurvature(whicharehyperbolicrelativetonilpotentcuspsubgroups)[26]and
limitgroups(whicharehyperbolicrelativetomaximalnon-cyclicabeliansubgroups)
[1],[21]. Exactnessoflimitgroupsalsofollowsfromtheirlinearity.
Anotherinterestingcaseofgroupactionswhichimpliesexactnessisaproperand
co-compactactiononafinitedimensionalCAT(0)cubicalcomplex[12].
Themappingclassgroup(cid:2)(S)ofacompactorientablesurfaceS isalsoanatural
example of an exact group [42], [49]. Indeed, the action of (cid:2)(S) on the space of
complete geodesic laminations is amenable [42]. In contrast, the more well-known
actionof(cid:2)(S)ontheThurstonboundaryPMF ofTeichmüllerspaceisnotamenable
because of non-amenable isotropy subgroups. However, if we denote by K the set
ofallnon-trivialisotopyclassesofnon-peripheralsimpleclosedcurvesonS (i.e.,K
is the vertex set of the curve complex of S), then the assumption of Proposition 2.7
is satisfied for X = PMF [49]. Since every isotropy subgroup of a point in K is
a mapping class group of lower complexity, induction applies and the exactness of
(cid:2)(S)follows.
Sofarwehaveenumeratedexamplesofexactgroupsasmanyaswecan(theauthor
is sorry for any possible omission). Unfortunately, there does exist a (finitely pre-
sented)groupwhichisneitherexactnorcoarselyembeddableintoaHilbertspace[37].
Currently,itisnotknownwhetherthefollowinggroupsareexactornot: Thompson’s
groupF,Out(F ),automaticgroups,3-manifoldgroups,groupsofhomeomorphisms
r
(resp.diffeomorphisms)on(say)thecircleS1,(free)Burnsidegroupsandothermon-
strousgroups.
The rest of this section is devoted to the relationship of exactness to operator
∗ ∗
algebras. Associated with a group, there are the reduced group C -algebra C ((cid:2))
λ
andthegroupvonNeumannalgebraL((cid:2)). When(cid:2)isabelian,C∗((cid:2))isisomorphic
toC((cid:7)(cid:2)), whileL((cid:2))isisomorphictoL∞((cid:7)(cid:2)), where(cid:7)(cid:2) isthePoλntrjagindualof(cid:2).
∗
Hence the study of C ((cid:2)) corresponds to “noncommutative topology” and that of
λ
L((cid:2))to“noncommutativemeasuretheory”[15]. Amenabilityof(cid:2)canbereadfrom
itsoperatoralgebras.
Theorem2.11([41],[54],[74]). Foragroup(cid:2),thefollowingareequivalent.
1. Thegroup(cid:2) isamenable.
∗ ∗
2. ThereducedgroupC -algebraC ((cid:2))isnuclear.
λ
Amenableactionsandapplications 1571
3. ThegroupvonNeumannalgebraL((cid:2))isinjective.
Ageneralizationofthistheoremtoagroupactiongoesasfollows.
Theorem2.12([3]). Fora(compact)(cid:2)-spaceX,thefollowingareequivalent.
1. The(cid:2)-spaceXisamenable.
2. ThereducedcrossedproductC∗-algebraC∗(X(cid:2)(cid:2))isnuclear.
λ
3. Thegroup-measure-spacevonNeumannalgebraL(X(cid:2)(cid:2),m)isinjectivefor
any(cid:2)-quasi-invariantRadonprobabilitymeasuremonX.
∗ ∗
ThenuclearC -algebrasareaccessibleamongtheC -algebrasandtheclassifica-
∗ ∗
tion program of nuclear C -algebras is a very active area of research in C -algebra
theory [73]. Many C∗-algebras C∗(X(cid:2)(cid:2)) arising from various kinds of boundary
λ
actions are classifiable via their K-theory [4], [53], [77]. Unlike the group case, a
∗ ∗
C -subalgebraofanuclearC -algebraneedsnotbenuclear. Thenotionofexactness
was introduced to give an abstract characterization of subnuclearity and has met a
greatsuccess[50],[51]. Exactnesshasadeepconnectionwithoperatorspacetheory
∗
[51],[69]. AC -algebraAiscalledexact iftakingtheminimaltensorproductwith
∗
A preserves short exact sequences of C -algebras. The following theorem explains
thenomenclatureofexactgroups.
Theorem2.13([11],[40],[51],[60]). Foragroup(cid:2) thefollowingareequivalent.
1. Thegroup(cid:2) isexact.
∗ ∗
2. ThereducedgroupC -algebraC ((cid:2))isexact.
λ
3. ThegroupvonNeumannalgebraL((cid:2))isweaklyexact.
∗ ∗
We note that a C -subalgebra of an exact C -algebra is always exact and that a
von Neumann subalgebra of a weakly exact von Neumann algebra is weakly exact
provided that there exists a normal conditional expectation. Since a von Neumann
∗
algebrawithaweaklydenseexactC -algebraisweaklyexact,weobtainthefollowing
corollary.
Corollary2.14. Exactnessisclosedundermeasureequivalence.
We recall that two groups (cid:2) and (cid:3) are measure equivalent [36] if there exist
commutingmeasurepreservingfreeactionsof(cid:2) and(cid:3)onsomeLebesguemeasure
space((cid:14),m)suchthattheactionofeachofthegroupsadmitsafinitemeasurefunda-
mentaldomain. Forexample,latticesinthesame(secondcountable)locallycompact
groupGaremeasureequivalent. Itisknownthatmeasureequivalencecoincideswith
thestableorbitequivalence[29]andhencegivesrisetoastableisomorphismofthe
correspondinggroup-measure-spacevonNeumannalgebras.
1572 NarutakaOzawa
3. Amenablecompactificationswhicharesmall
We study the “size” of an amenable compactification with its application to von
Neumannalgebratheoryinmind. Acompactificationofagroup(cid:2)isacompactspace
(cid:15)(cid:2)containing(cid:2)asanopendensesubset. Weonlyconsiderthosecompactifications
whichareequivariant;theleftmultiplicationactionof(cid:2)on(cid:2)extendstoacontinuous
action of (cid:2) on (cid:15)(cid:2). A group (cid:2) is amenable iff the one-point compactification is
amenable,andagroup(cid:2)isexactifftheStone–Cˇechcompactificationβ(cid:2)isamenable.
Thus we think that the “size” of an amenable compactification of a given group
measures the “degree of amenability” of the group. We say that a compactification
(cid:15)(cid:2) of (cid:2) is small at infinity if for every net (s ) in (cid:2) with s → x ∈ ∂(cid:2), we have
n n
s t → x for every t ∈ (cid:2) [13]. In other words, (cid:15)(cid:2) is small at infinity if every flow
n
in (cid:2) drives (cid:2) to a single point. We note that (cid:15)(cid:2) is small at infinity iff the right
multiplicationactionof(cid:2)extendscontinuouslyon(cid:15)(cid:2)insuchawaythatitistrivial
on (cid:15)(cid:2) \(cid:2). For instance, the Gromov compactification F ∪∂F of the free group
r r
F (cf.Example2.2)issmallatinfinitysincethefirstksegmentofst issameasthat
r
ofs aslongas|s|≥k+|t|. Thesameappliestogeneralhyperbolicgroups.
Wesaythatagroup(cid:2)belongstotheclassSifthecompact((cid:2)×(cid:2))-space∂β(cid:2) =
β(cid:2)\(cid:2)(withthebilateralaction)isamenable. If(cid:2)hasanamenablecompactification
(cid:15)(cid:2) which is small at infinity, then we have (cid:2) ∈ S. It follows that the class S
containsamenablegroupsandhyperbolicgroups(ormoregenerally,anygroupwhich
is hyperbolic relative to a family of amenable subgroups). The class S is closed
undersubgroupsandfreeproducts(withfiniteamalgamations). Moreover,thewreath
product(cid:3)(cid:20)(cid:2)ofanamenablegroup(cid:3)byagroup(cid:2) ∈SagainbelongstoS[62]. We
observethataninneramenablegroupinShastobeamenablebecause,bydefinition,
agroup(cid:2)isinneramenableif∂β(cid:2)carriesaninvariantRadonprobabilitymeasurefor
theconjugationactionof(cid:2)(cf.[44]). Ingeneral,foragivengroup(cid:2)andacountable
(cid:2)-spaceK,itisaninterestingproblemtodecidewhetherornotthecompact(cid:2)-space
∂βK = βK \K is amenable (cf. [62]). We note the trivial case where the isotropy
subgroupsareallamenable.
Thefollowingisarelativeversionofsmallness.
Definition3.1. LetGbeanon-emptyfamilyofsubgroupsof(cid:2). Foranet(s )in(cid:2)
n
we say that s → ∞ relative to G if s ∈/ s(cid:3)t for any s,t ∈ (cid:2) and (cid:3) ∈ G. We
n n
say that a compactification (cid:15)(cid:2) is small relative to G if for every net (s ) in (cid:2) with
n
s →x ∈(cid:15)(cid:2) andwiths →∞relativetoG,wehaves t →x foreveryt ∈(cid:2).
n n n
Supposethatagroup(cid:2)actsonasimplicialtreeT andletT bethecompactification
definedintheprevioussection. Werecallthattheopenbasisofthetopologyisgiven
bycuttingfinitelymanyedges. Wefixabasepointe ∈ T andconsiderthesmallest
compactification (cid:15)T(cid:2) of (cid:2) for which the map (cid:2) (cid:3) s (cid:4)→ s.e ∈ T is continuous on
(cid:15)T(cid:2). Then(cid:15)T(cid:2) issmallrelativetothefamilyofedgestabilizers. Indeed,suppose
thats →∞relativetoedgestabilizersandthata,b ∈T aregiven. Letγ beapath
n
connecting a to b. Since the net (s .γ) of paths leaves every edge, two end points
n