COHOMOLOGICAL FINITENESS CONDITIONS FOR ELEMENTARY AMENABLE GROUPS For ...

14 Pages·2008·0.2 MB·English

COHOMOLOGICAL FINITENESS CONDITIONS FOR ELEMENTARY
AMENABLE GROUPS
P.H.KROPHOLLER,C.MARTÍNEZ-PÉREZ,ANDB.E.A.NUCINKIS
ABSTRACT. It is proved that every elementary amenable group of type FP∞
admitsacocompactclassifyingspaceforproperactions.
1. INTRODUCTION
For discrete groups the term cohomological finiteness condition refers to any
propertyofgroupswhichholdsforallgroupsthatadmitfiniteEilenberg–MacLane
spaces. Amongst such properties there are classical finiteness conditions such as
finite generation and finite presentability and there are also the more exotic con-
ditions such as type FP , type FP, and type FL. Note also that the property of
∞
beingtorsion-freeisacohomologicalfinitenessconditionbutnotaclassicalfinite-
nessconditionwhileresidualfinitenessisaclassicalfinitenessconditionbutnota
cohomologicalfinitenesscondition.
In recent years there has been increasing interest in a variant of the Eilenberg–
MacLanespace,namelytheclassifyingspaceforproperactions. Inthispaperwe
shall determine the precise conditions under which elementary amenable groups
admit cocompact proper classifying spaces. Bredon cohomology plays a role in
studying these classifying spaces in a way that runs largely parallel to the role of
ordinarygroupcohomologyinstudyingEilenberg–MacLanespacesandtheiruni-
versalcovers. InBredoncohomology,thegroupGisreplacedbytheorbitcategory
OXGdefinedwithrespecttoasuitablefamilyofsubgroupsX. Inthispaperwe
shall only be concerned with the family F of finite subgroups and so we simply
writeOGfortheorbitcategoryinsteadofOFG. Modulesovertheorbitcategory
arecontravariantfunctorstothecategoryofabeliangroups. ThesearecalledOG-
modules. The category of OG-modules has enough projectives and homological
algebra can be developed using projective resolutions giving rise to the Bredon
cohomologyofgroupswithclearresemblancetoordinarygroupcohomology.
AllthenotionstypeFP ,typeFP,typeFL,etchaveanalogueswhenconsidering
∞
BredonprojectiveresolutionsandwerefertothesebythenamestypeBredonFP ,
∞
typeBredonFP,typeBredonFL,etc. Figure1showsthevariousinterrelationships
2000MathematicsSubjectClassification. 57Q05,20J05.
Key words and phrases. Bredon cohomology, elementary amenable group, proper classifying
space,Eilenberg–MacLanespace,finitetype.
ThesecondnamedauthorwaspartiallysupportedbyMTM2004-08219-C02-01andGobiernode
Aragón.
2 P.H.KROPHOLLER,C.MARTÍNEZ-PÉREZ,ANDB.E.A.NUCINKIS
α (cid:37)(cid:57) ∗ (cid:37)(cid:57) β (cid:37)(cid:57)
Bred(cid:69)(cid:89)on F Bredo(cid:69)(cid:89)n FL Bredo(cid:69)(cid:89)n FP BredonFP∞
γ γ γ δ
(cid:5)(cid:25)
α (cid:37)(cid:57) ∗ (cid:37)(cid:57) β (cid:37)(cid:57)
ε F FL FP FP∞
β ε
(cid:20)(cid:40) (cid:5)(cid:25) (cid:5)(cid:25)
finitelypresented torsion-free (cid:42)(cid:62) FP
2
α
FIGURE 1. Connectionsbetweencohomologicalfinitenessconditions
betweenthesepropertiesandalltheseimplicationsareeasilyestablished. Theno-
tion of type Bredon F is by definition equivalent to the existence of a cocompact
proper classifying space. Save for the two implications marked ∗ all the implica-
tionsshownareknowntobeirreversible. Wehaveincludedashortexplanationof
thisdiagraminthelastsectionofthispaper.
For virtually soluble groups or, more generally, elementary amenable groups
therearenosurprisesastowhichgroupsareofBredontypeFandourmaintheo-
remisasfollows.
Theorem 1.1. Let G be an elementary amenable group. Then the following are
equivalent.
(i) GisoftypeBredonF.
(ii) GisoftypeBredonFL.
(iii) GisoftypeBredonFP.
(iv) GisoftypeBredonFP .
∞
(v) GisvirtuallyoftypeF.
(vi) GisvirtuallyoftypeFL.
(vii) GisvirtuallyoftypeFP.
(viii) GisoftypeFP .
∞
(ix) Gisconstructible.
(x) Either G is polycyclic-by-finite or G has a normal subgroup K such that
G/K isaEuclideancrystallographicgroupandforeachsubgroupL⊇K
with L/K finite there is a finitely generated virtually nilpotent subgroup
B = B(L) of L and an element t =t(L) of L such that t−1Bt ⊂ B and
L=B∗ is a strictly ascending HNN-extension with base B and stable
B,t
lettert.
Here(ix)referstothenotionofconstructiblegroupasintroducedbyBaumslag
and Bieri [3]. The class of constructible groups is the smallest class of groups
closed under forming amalgamated free products, HNN-extensions and finite ex-
tensionsandallsuchgroupsarefinitelypresentedandoftypeFP . Baumslagand
∞
Bieri provide a thorough discussion of the nature of soluble constructible groups,
[3]. Furtheranalysisofthisclassappearsinthework[11]ofBieriandStrebeland
playsacrucialroleinthispaper.
In view of known interconnections between the conditions (i)–(x), which we
discussbelow, mostoftheworkinthispaperisconcernedwithestablishing(viii)
COHOMOLOGICALFINITENESSCONDITIONSFORELEMENTARYAMENABLEGROUPS 3
⇒ (x) ⇒ (i). The first of these implications is the subject of §2 and the second is
the subject of §4. For the second implication we use Bredon cohomology which
wereviewin§3.
The following is an immediate consequence. Note that prior to this work this
CorollarywasnotprovenevenfortheclassofsolublegroupsoftypeFP.
Corollary1.2. EveryelementaryamenablegroupoftypeFPisoftypeF.
Proof. LetGbeanelementaryamenablegroupoftypeFP. AccordingtoTheorem
1.1, G is of type Bredon F and so admits a cocompact model for the classifying
space for proper actions EG. Since every group of type FP, in particular G, is
torsionfreeitfollowsthatEG=EGandGisoftypeF. (cid:3)
Corollary 1.3. Let G be an elementary amenable group of type Bredon F and let
ΦbeafinitegroupofautomorphismsofG. ThenthesubgroupFix(Φ)ofelements
ofGfixedbyeveryelementofΦisalsooftypeBredonF.
Proof. The split extension, or semidirect product, G(cid:111)Φ is elementary amenable
and of type FP since both of these properties are inherited by finite index over
∞
groups. Therefore, according to Theorem 1.1 (viii) ⇒ (i), G(cid:111)Φ is of type F.
NormalizersoffinitesubgroupsofgroupsoftypeFarealwaysoftypeFandhence
thenormalizerNG(cid:111)Φ(Φ)isoftypeF. ThesubgroupFix(Φ)hasfiniteindexinthis
normalizerandsoisalsooftypeF. (cid:3)
ForanygroupGsatisfyingtheconditionsintheTheoremonehas
hG=cdQG=cdG<∞
where hG is the Hirsh rank of G and cdG the Bredon cohomological dimension,
which is the analogous of the usual cohomological dimension but for Bredon co-
homology. Thisleadstothefollowing:
Conjecture1.4. Theconditions
(xi) hG=cdQG<∞
(xii) hG=cdG<∞
canbeaddedtotheTheorem.
It follows from [12] that for any group cdQG≤cdG and hence (xii) implies (xi)
in the above conjecture. Furthermore (xi) implies that G/T is of type Bredon F,
whereT isthelargestnormallocallyfinitesubgroupofG. Henceapositiveanswer
tothefollowingwouldalsoproveConjecture1.4.
Conjecture1.5. LetGbeanelementaryamenablegroupsuchthathG=cdQGis
finite. ThenGhasaboundontheordersofthefinitesubgroups.
Background to the Theorem. Historically, the very first steps towards under-
standing cohomological finiteness conditions for soluble groups were taken by
GruenbergandStammbach. KeystepsconcerningnilpotentgroupsappearinGru-
enberg’snotes[18]andhomologicaldimensionwascomputedbyStammbach[32].
ThequestionwasfurtherinvestigatedandhighlightedbyBieri[6]. Gildenhuysde-
termined exactly which soluble groups have cohomological dimension 2: the so-
lution[15]showsthatallnon-abeliansuchgroupsareascendingHNN-extensions
of type F and is the first evidence that questions about cohomological finiteness
conditions for soluble groups would prove to be substantial and interesting. For
4 P.H.KROPHOLLER,C.MARTÍNEZ-PÉREZ,ANDB.E.A.NUCINKIS
soluble-by-finite groups, the equivalence of conditions (vi), (vii) and (ix) of The-
orem 1.1 was established by Gildenhuys, Strebel and Kropholler, [16, 17, 20].
Subsequentlyitwasshown[21,22]byKrophollerthatsolublegroupsoftypeFP
∞
are virtually of type FP and work [19] of Hillman and Linnell made it possible
to extend the results to the elementary amenable case. At this stage it became
clearthateveryelementaryamenablegroupoftypeFP isnilpotent-by-abelian-by-
∞
finite,constructibleandvirtuallyoftypeF. Howeveritremainedanopenproblem
whetherornotelementaryamenablegroups,orevensolublegroups,oftypeFPare
necessarily of type FL. Moreover, the interest in proper classifying spaces, which
are natural to consider for groups with torsion, raised questions as to whether all
solublegroupsoftypeFP satisfiedthestrongestBredonfinitenessconditions,see
∞
forexample[27]wheretheequivalencebetween(viii)and(iii)isproven. Theorem
1.1 shows that this is the case. Part (x) comes about through a careful analysis of
theBieri–Strebelstrategyforcharacterizingpropertiesofnilpotent-by-abelian-by-
finitegroupsusinginvariants[11],commonlycalledBNS-invariants[7],whichare
subsetsofcertainvaluationspheres.
2. BIERI–STREBEL INVARIANTS FOR NILPOTENT-BY-ABELIAN-BY-FINITE
GROUPS, AND THE PROOF OF THEOREM 1.1 (viii)⇒(x)
The goal of this section is to establish the more refined structure theory for G
statedinTheorem1.1(x). Asexplainedabove,resultsin[21,22]andin[19]imply
thatanyelementaryamenablegroupoftypeFP isfinite-by-virtuallysoluble. By
∞
takingthecentralizerofthefinitenormalsubgroup,whichissolubleoffiniteindex
oneeasilyseesthatthegroupisinfactnilpotent-by-abelian-by-finite.
WeadoptthenotationusedbyBieriandStrebelin[11]. Forafinitelygenerated
abeliangroupQwewriteS(Q)forthevaluationsphereasdefinedin§1.1of[11].
Let (N,H) be an admissible pair of subgroups of the group G, meaning that the
followingconditionsaresatisfied:
• N andH arebothnormalsubgroupsofG;
• N ⊆H;
• N isnilpotent;
• H/N isabelian;and
• G/H isfinite.
NowletPdenotethelargestnormallocallypolycyclicsubgroupofH. ThenH/P
is a finitely generated abelian group and so the valuation sphere S(H/P) is de-
fined. Moreover, BieriandStrebelshowthatif(N(cid:48),H(cid:48))isanotheradmissiblepair
then the valuations spheres S(H/P) and S(H(cid:48)/P(cid:48)) can be identified in a canonical
way. ThereforetheydefinethevaluationsphereS(G)tobeS(H/P)forsomefixed
choiceofadmissiblepair(N,H)withoutanyessentialambiguity.
BieriandStrebelintroducetheinvariantσ(G), acertainclosedsubsetofS(G).
We shall be interested in the following results about this invariant which are the
contentofTheorems5.2and5.4of[11].
Proposition2.1.
(i) Gisconstructibleifandonlyifσ(G)iscontainedinanopenhemisphere.
(ii) Gispolycyclic-by-finiteifandonlyifσ(G)isempty.
TakingQtobethequotientgroupH/N wenaturallyhavethatS(G)=S(H/P)
isasubsphereofS(Q).
COHOMOLOGICALFINITENESSCONDITIONSFORELEMENTARYAMENABLEGROUPS 5
WenownotethattheactionofGbyconjugationonH stabilizesσ(G)(see[27,
Lemma3.4]). MoreoveritinducesanactionofthefinitegroupG/H onbothH/N
and H/P. In turn this induces actions of G/H on the vector spaces hom(H/P,R)
and hom(H/N,R) and hence also on the valuation spheres S(G) = S(H/P) and
S(H/N)stabilizingσ(G).
Also,thegroupGactsbyconjugationonN andthispassestoanactionofG/N
onthelargestabelianquotientN =N/[N,N]ofN. InthiswaywemayviewN
ab ab
as a right Z[G/N]-module. For x ∈ G and b ∈ N we write bx for the conjugate
x−1bxandwewritexfortheimageofxinG/N,thatisxisthecosetNx. Ifaisan
elementofN sothatisa=[N,N]bforsomeb∈N thenwewriteaxforthecoset
ab
[N,N]bx. InotherwordsweshallnotateN asarightG/N-module.
ab
Q ofQandafinitelygeneratedsubgroupB ofN
0 ab
Associated to the data G, H, N we may consider the following subsets B and
0
B ofG.
• B isdefinedtobethesetofx inGforwhichthereexistsafinitelygen-
0
eratedsubgroupB ofN suchthatB x⊆B and (cid:91)B xi=N .
0 ab 0 0 0 ab
i≤0
• B isdefinedtobethesetofxinGforwhichthereexistsafinitelygener-
atedsubgroupBofN suchthatBx⊆Band (cid:91)Bxi =N.
i≤0
WeshallnowmaketheassumptionthatGisconstructible. ThereforebyPropo-
sition 2.1 (i) the invariant σ(G) is contained in an open hemisphere of S(G) and
the following three lemmas all rely on this. We shall also assume that σ(G) is
non-empty,i.e.,thatGisnotpolycyclic-by-finite.
Lemma 2.2. There exists x ∈ B whose image x in G/N belongs to the centre
0
ζ(G/N).
Proof. AsremarkedabovethefinitegroupG/H actsonQandonthesphereS(Q)
stabilizingσ(G). Nowlet
C/N =C (G/H)≤ζ(G/N).
H/N
Since Q is finitely generated abelian [27, Lemma 3.5] implies that by changing
H if necessary to a finite index subgroup we may assume that H/N is torsion
free and H/N =C/N×T/N with T/N as in [27, Lemma 3.6]. So applying that
resultwededucethatσ(C)iscontainedinanopenhemisphere. IfrkC/N=0then
σ(C)=∅, and henceC and also G would be polycyclic-by-finite. Therefore by
[10, Theorem 4.6] there are elements q ,...,q of C/N and a finitely generated
1 s
subgroup B of N such that q all satisfy the condition B q ⊆ B and N =
0 ab i 0 i 0 ab
(cid:91)B (q ...q )i. Itsufficesthentotakexwithx¯=q ...q . (cid:3)
0 1 n 1 s
i<0
The previous result can also be proven as follows. Since σ(G) is non empty,
compact (in fact results of Bieri and Strebel show that it is finite in this case) and
stabilizedbythefinitegroupG/H itfollowsthatitscentreofmassisafixedpoint
ofG/H. Thenonecanargueinasimilarwayasin[10,Theorem4.6]butusingthis
fixedelementtodeducetheexistenceofaafinitesubset{q ,...,q }ofQwiththe
1 m
followingproperties:
• {q ,...,q } is invariant under the action of the finite group G/H, and
1 m
(cid:104)q ,...,q (cid:105)generateasubgroupoffiniteindexinQ.
1 m
6 P.H.KROPHOLLER,C.MARTÍNEZ-PÉREZ,ANDB.E.A.NUCINKIS
• ThereisafinitelygeneratedsubgroupB ofN suchthattheq allsatisfy
0 ab i
theconditionB q ⊆B .
0 i 0
• N = (cid:91)B (q ...q )i.
ab 0 1 m
i<0
Thenonetakesx∈Gwithx¯=q ...q .
1 m
Lemma2.3. B =B.
0
Proof. Thisisaneasyvariationontheproofof([11],Theorem5.2). Takex∈B
0
andconsiderthederivedseriesofN with
γ N =N,
1
γ N =[γN,N].
i+1 i
We prove by induction on (cid:96) that there is some finitely generated subgroup B
(cid:96)+1
of N/γ N with Bx ⊆ B and (cid:91)Bxi = N. When (cid:96) = 1 there is nothing
(cid:96)+1 (cid:96)+1 (cid:96)+1 (cid:96)+1
i≤0
to prove and so we assume that (cid:96) ≥ 2 and that there exists a finitely generated
subgroup B with the desired properties. Let A be a finitely generated subgroup
(cid:96) (cid:96)
of N/γ N with A γ N/γ N =B . Exactly as in ([11], Theorem 5.2) one gets a
(cid:96)+1 (cid:96) (cid:96) (cid:96) (cid:96)
finitelygeneratedsubgroupA ofγ N/γ N havingsimilarpropertiesasthoseof
0 (cid:96) (cid:96)+1
B anditsuficestotake
(cid:96)
B =A A .
(cid:96)+1 (cid:96) 0
(cid:3)
Using Lemma 2.2, choose x∈B such that x belongs to ζ(G/N). By Lemma
0
2.3, x belongs to B and so we may choose a finitely generated subgroup B of N
suchthatBx ⊆BandN = (cid:91)Bxi. WenowkeepxandBfixedfortheremainderof
i<0
thissection. NotethatifxhasfiniteorderthenGispolycyclic-by-finite. Weshall
thereforeassumethatxhasinfiniteorder.
Lemma 2.4. The subgroup K :=(cid:104)B,x(cid:105) is normal in G and if L is any subgroup
containingK suchthatL/K isfinitethenthereexisty∈Gandafinitelygenerated
subgroupDofLsuchthat
(i) BisasubgroupoffiniteindexinD,
(ii) Dy⊆D,
(iii) yk =x(cid:96) forsomepositiveintegersk,(cid:96),
(iv) L=(cid:104)D,y(cid:105).
Proof. Since N = (cid:91)Bxi we have that N ⊆K and clearly K/N is the cyclic group
i<0
generatedbyxwhichiscentralinG/N. ThusK isnormalinG.
Suppose now that L is a subgroup of G containing K such that L/K is finite.
Then L/N is virtually infinite cyclic and centre-by-finite. Let N /N denote the
1
largest finite normal subgroup of L/N. Then L/N is infinite cyclic. Choose y
1
to be a generator of L modulo N . Then y and x generate commensurable cyclic
1
subgroups of L/N and so, replacing y by y−1 if necessary, we may assume that
there are positive integers k and (cid:96) such that yk =x(cid:96). Notice that x(cid:96)y−k is then an
elementofN.
LetN beafinitesubsetofGconsistingofcosetrepresentativesfortheelements
1
of N /N. Since x is central in G/N it follows that for each g∈N there exists
1 1
COHOMOLOGICALFINITENESSCONDITIONSFORELEMENTARYAMENABLEGROUPS 7
n ∈N such that gx =gn . Choose j<0 so that Bxj contains all the elements n
g g g
asgrunsthroughN andsothatitalsocontainsx(cid:96)y−k. Thechoiceof jispossible
1
because there are only finite many n and N is the directed union (cid:91)Bxi. Now
g
i<0
consider the group B :=(cid:104)Bxj ∪N (cid:105). This is contained in the virtually nilpotent
1 1
groupN andsoisitselfvirtuallynilpotent. Also,B isfinitelygenerated,Bx ⊆B ,
1 1 1 1
andwehaveN = (cid:91)Bxi. DefineDby
1 1
i<0
y yk−1
D:=(cid:104)B ,B ,...,B (cid:105).
1 1 1
Then D has the desired properties. First, Dy is generated by By,...,Byk−1,Byk and
1 1 1
sincex(cid:96)y−k belongstoB wehaveByk =Bx(cid:96) ⊆B sothatDy⊆D. Secondly(cid:104)D,y(cid:105)
1 1 1 1
containsN,N andysoequalsL. (cid:3)
1
Establishing the structure described in Theorem 1.1(x). There are two cases
according to whether or not σ(G) is empty. If σ(G)=∅ then G is polycyclic-
by-finite by Proposition 2.1 (ii) and we are done. If σ(G)(cid:54)=∅ then Lemma 2.4
applies. Inthatcase,letKbeasinLemma2.4andletK /Kbelargestfinitenormal
1
subgroup of G/K. Since G/K is a quotient of G/H it is finitely generated and
abelian-by-finite. Therefore G/K is a Euclidean crystallographic group. Lemma
1
2.4showsthatanyovergroupLoffiniteindexoverKorK alsoenjoysthestructure
1
of being an ascending HNN-extension. Therefore we may replace K by K and
1
havethedesiredconclusion.
3. BREDON COHOMOLOGY AND FINITENESS CONDITIONS FOR PROPER
CLASSIFYING SPACES
Let G be a group. We write OG for the orbit category of G with respect to
theclassoffinitesubgroupsofG. TheorbitcategoryhasthetransitiveG-setswith
finitestabilizersasobjectsandG-mapsbetweenthemasmorphisms. Modulesover
theorbitcategoryarecontravariantfunctorsfromtheorbitcategorytothecategory
ofabeliangroups. AsequenceA→B→CofOG-modulesisexactatBifandonly
if each instance is exact, that is A(∆)→B(∆)→C(∆) is exact at B(∆) for every
transitiveG-set∆.
ForG-sets∆,Ωwewrite[∆,Ω] forthesetofG-mapsfrom∆toΩandwewrite
G
Z[∆,Ω] forthefreeabeliangroupon[∆,Ω] . Whenthereisnoambiguitywedrop
G G
thesymbolGandsimplywrite[∆,Ω]. FixingΩandallowing∆torangeovertran-
sitive G-sets with finite stabilizers we obtain an OG-module Z[ ,Ω]. The trivial
OG-module,usuallywrittenZ,arisesfromthisconstructionbytakingΩtobethe
one-point G-set. For any finite group H, the OG-module Z[ ,H\G] is projective
and direct sums of modules of this form (allowing different finite subgroups) are
called free OG-modules. Every projective OG-module is a direct summand of a
free module and the finitely generated projective modules are precisely the direct
summands of finite direct sums of modules of the form Z[ ,H\G] with H finite.
The notions of type Bredon FP, Bredon FL, Bredon FP are defined in terms of
∞
projectiveresolutionsofZoverOGinjustthesamewaythattheclassicalnotions
oftypeFP,FLandFP aredefined.
∞
8 P.H.KROPHOLLER,C.MARTÍNEZ-PÉREZ,ANDB.E.A.NUCINKIS
Moreover it is also possible to define the notion of type Bredon FP for each
n
n≥0. ThefollowingLemmaimpliestheBredonanalogueoftheclassicalfactthat
agroupisoftypeFP ifandonlyifitisfinitelygenerated.
1
Lemma3.1. LetGbeagroup. ThenGisoftypeBredonFP ifandonlyifGhas
0
only finitely many conjugacy classes of finite subgroups, and G is of type Bredon
FP ifandonlyif,inaddition,theWeyl-groupK\N (K)ofeachfinitesubgroupK
n G
isoftypeFP .
n
Proof. IfGisoftypeBredonFP thenthereisaG-finiteG-setΩwithfinitestabi-
0
lizersandanepimorphism
Z[ ,Ω]→Z.
Now let K be an arbitrary finite subgroup of G. Evaluating this epimorphism at
Ω we obtain an epimorphism ZΩK → Z and therefore ΩK is non-empty. This
showsthatKbelongstothefinitesetofconjugacyclassesofsubgroupswhichhave
fixedpointsinΩ. Conversely, ifthereareonlyfinitelymanyconjugacyclassesof
(cid:71)
finite subgroups then we can take Ω = H\G where H runs through a set of
H
conjugacyclassrepresentativesoffinitesubgroups,andtheobviousaugmentation
mapZ[ ,Ω]→Zisanepimorphism.
The necessary and sufficient conditions for Bredon type FP are consequences
n
ofthefollowinglemma. (cid:3)
We say a Bredon module M is finitely generated if there is a finite OG-set Σ in
the sense of Lück [24, 9.16, 9.19] such that there is a free Bredon module F on Σ
mapping onto M. An OG-set Σ is determined by sets Σ for each finite subgroup
K
K. ΣissaidtobefiniteifforallfinitesubgroupsK,Σ isfiniteandΣ =∅forall
K H
butfinitelymanyfinitesubgroupsH.
Lemma 3.2. Let G be a group with finitely many conjugacy classes of finite sub-
groups. Then a Bredon module M is of type Bredon FP if and only if for each
n
finite subgroup K of G, M(K\G) is a module of type FP over the Weyl-group
n
WK =K\N (K).
G
Proof. Let M be a Bredon module of type Bredon FP and let P (cid:16)M be a pro-
n ∗
jective resolution. We may assume that all P for i≤n are finitely generated free
i
Bredonmodules. Anargumentanalogousto[29,Section3]showsthatuponevalu-
atingtheP(K\G)arefinitelygeneratedpermutationmodulesoverWK withfinite
i
stabilizers. Hence, by [28, Proposition 6.3], P(K\G) is of type FP for all i≤n.
i ∞
Adimensionshift,see[6,Proposition1.4],impliesthatM(K\G)isaWK-module
oftypeFP .
n
The converse is proved by induction on n. Let n = 0 and M be a Bredon mod-
ule such that M(K\G) is a finitely generatedWK-module for all finite subgroups
K. We will construct a finite OG-set Σ generating M as a Bredon module. The
free Bredon module on Σ then is finitely generated and maps onto M. Recall
that for every G-map H\G → K\G there is a homomorphism of abelian groups
ϕH :M(K\G)→M(H\G).ForeachfinitesubgroupH ofGfixafinitegenerating
K
set X of M(H\G). Now Σ is the union of X with all elements of M(H\G) of
H H H
the form ϕH(x ) whenever there is a G-map H\G → K\G and x ∈ X . Since
K K K K
there are only finitely many conjugacy classes of finite subgroups this results in a
finite set Σ . Since only one representative for each finite subgroup needs to be
H
COHOMOLOGICALFINITENESSCONDITIONSFORELEMENTARYAMENABLEGROUPS 9
takenintoaccount,theresultingOG-setisindeedfinite. TherearemapsΣ →Σ
K H
inducedbythemapsϕH andthefreemoduleF onΣmapsontoM.
K
Nowsupposen>0andtheclaimistruefork<n.Sinceforeachfinitesubgroup
M(K\G)isaWK-moduleoftypeFP itisinparticularfinitelygenerated. Andwe
n
haveshownthatthereisashortexactsequenceofBredonmodulesK (cid:26)P (cid:16)M
0 0
withP finitelygeneratedfree. Then,asabove,P (H\G)isaWH-moduleoftype
0 0
FP . Hence by [6, Proposition 1.4] all K (H\G) areWH-modules of type FP
∞ 0 n−1
for all finite subgroups H of G. By induction K is of type Bredon FP and the
0 n−1
claimfollows. (cid:3)
Let X be a G-complex in the sense of tom Dieck ([34], Chapter II): this is a
G-CW-complexonwhichGactsbypermutingthecellsandinsuchawaythatthe
stabilizer of each cell fixes that cell point by point. We shall write ∆n(X) for the
setofn-cellsofX.
Lemma 3.3. Let X be a G-complex such that the fixed sets XH are acyclic for all
finitesubgroupsofG. ThentheaugmentedBredoncellcomplex
···→Z[ ,∆n(X)]→Z[ ,∆n−1(X)]→···→Z[ ,∆1(X)]→Z[ ,∆0(X)]→Z→0
isanexactsequenceofOG-modules.
Proof. To check that the Bredon cell complex is exact it suffices to check that the
chaincomplexobtainedbyevaluatingoneachtransitiveG-setwithfinitestabilizers
is exact. A typical such G-set has the form H\G, that is the set of right cosets of
somefinitesubgroupH. ForanyG-setΩtheset[H\G,Ω]canbeidentifiedwiththe
H-fixedpointsetΩH andsowhenweevaluatetheBredoncomplexatH\G,what
we see is the ordinary augmented cellular chain complex of the space XH. Thus
thelemmafollowsfromtheassumptionthatallthefixedsetsXH areacyclic. (cid:3)
Corollary3.4. IfT isaG-treewithedgesetE andvertexsetV thentheaugmented
Bredoncellcomplex
0→Z[ ,E]→Z[ ,V]→Z→0
isashortexactsequenceofOG-modules.
Proof. If H is any finite subgroup of G then the fixed set TH is again a tree and
henceacyclic. ThusthecorollaryfollowsfromLemma3.3. (cid:3)
Lemma3.5. LetGbeagroupandletBbeasubgroup. IfBisoftypeBredonFP
thenZ[ ,B\G]isanOG-moduleoftypeBredonFP.
Proof. This follows in much the same way as the corresponding result for groups
oftypeFP. First,thereisafunctor× G:OB→OG. Thisfacilitatesarestriction
B
functorfromthecategoryofOG-modulestothecategoryofOB-modulesandthe
restriction functor has, in turn, a left adjoint called induction. We need to know
that the induction functor is exact, that it carries finitely generated projective OB-
modules to finitely generated projective OG-modules, and that IndGZ[−,Z] ∼=
B B
Z[−,B\G] . Thelemmafollowsfromthesefactsbyapplyinginductiontoafinite
G
projectiveresolutionofthetrivialOB-moduleZ.
DetailscanbefoundinSymondsexposition[33]. Forthereader’sconvenience
weincludeasummaryofthekeysteps.
10 P.H.KROPHOLLER,C.MARTÍNEZ-PÉREZ,ANDB.E.A.NUCINKIS
The induction functor is defined in [24, 9.15] and takes the following form in
ournotation.
(IndGBM)(S\G)=M(−)⊗OBZ[S\G,−×BG]G
where M is a OB-module, S a finite subgroup of G and ⊗OB is the tensor product
definedforexamplein[24,9.12]. ForanyfinitesubgroupLofBthereisaYoneda-
typeformulaforthistensorproduct:
M(−)⊗OBZ[L\B,−]B=M(L\B).
Thereisabijection
[S\G,−× G] ∼= (cid:71) [Sx−1\B,−] ,
B G B
x∈(B\G)S
and,takingfreeabeliangroupsonbothsides,thisgivesrisetoanisomorphism
Z[S\G,−× G] ∼= (cid:77) Z[Sx−1\B,−] .
B G B
x∈(B\G)S
ThistogetherwiththeYonedaformulaaboveyield
(IndGM)(S\G)= (cid:77) M(Sx−1\B).
B
x∈(B\G)S
This formula is used in [33] to define induction. Note that as exactness means
exactness upon evaluation from the formula above one deduces that induction is
exact[33,2.9]. AndintheparticularcasewhenM=Z[ ,L\B]weget
IndGZ[ ,L\B] =Z[ ,L\G] .
B B G
ThisimpliesthatIndG takesfinitelygeneratedfreeOB-modulestofinitelygen-
B
erated free OG-modules (see [33, Lemma 2.9]), (which is not a surprise as the
restrictionfunctorisexact)andalsoimpliesthatIndGZ=Z[ ,B\G] (thisis[33,
B G
Lemma2.7]).
(cid:3)
We shall work with the Grothendieck group K (OG) of finitely generated pro-
0
jective OG-modules. If P is a finitely generated projective OG-module then we
write [P] for the corresponding class in the Grothendieck group. If M is an OG-
moduleoftypeBredonFPthenwewrite[M]fortheelement∑(−1)i[P]inK (OG)
i 0
i≥0
whereP →M isanychoiceoffiniteprojectiveresolutionofM overOG. Anap-
∗
plicationofSchanuel’slemmashowsthat[M]iswell-defined.
Lemma3.6. LetGbeagroupoftypeBredonFP. If[Z]=0inK (OG)thenGis
0
oftypeBredonFL.
Proof. Thisisaspecialcaseof[24,Theorem11.2a]. Moregenerallyonecanwork
withtheimageof[Z]inK(cid:101)0(OG)thequotientoftheGrothedieckgroupmodulothe
subgroupgeneratedbyclassesofBredonfreemodulesanditissufficienttocheck
vanishingthere. (cid:3)
Lemma3.7. Let0→A→B→C→0beashortexactsequenceofmodulesoftype
Bredon FP over OG. Then the equation [C]=[B]−[A] holds in the Grothendieck
groupK (OG).
0