When is Group Cohomology Finitary?
If $G$ is a group, then we say that the functor $H^n(G,-)$ is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with $n$th cohomology fu…
Authors: Martin Hamilton
WHEN IS GR OUP COHOMOLOGY FINIT AR Y? MAR TIN HAMIL TON Abstra t. If G is a group, then w e sa y that the funtor H n ( G, − ) is nitary if it omm utes with all ltered olimit systems of o ef- ien t mo dules. W e in v estigate groups with ohomolo gy almost everywher e nitary ; that is, groups with n th ohomology funtors nitary for all suien tly large n . W e establish suien t ondi- tions for a group G p ossessing a nite dimensional mo del for E G to ha v e ohomology almost ev erywhere nitary . W e also pro v e a stronger result for the sub lass of groups of nite virtual ohomo- logial dimension, and use this to answ er a question of Leary and Nuinkis. Finally , w e sho w that if G is a lo ally (p olyyli-b y- nite) group, then G has ohomology almost ev erywhere nitary if and only if G has nite virtual ohomologial dimension and the normalizer of ev ery non-trivial nite subgroup of G is nitely generated. 1. Intr odution Let G b e a group and n ∈ N . The n th ohomology of G is a funtor H n ( G, − ) := Ext n Z G ( Z , − ) from the ategory of Z G -mo dules to the ategory of Z -mo dules, and w e sa y that it is nitary if it omm utes with all ltered olimit systems of o eien t mo dules (see 3 . 18 in [1℄; also 6 . 5 in [14℄). Bro wn [4 ℄ has haraterised groups of t yp e FP ∞ in terms of nitary funtors (see also results of Bieri, Theorem 1 . 3 in [ 2℄): Prop osition 1.1. A gr oup G is of typ e FP ∞ if and only if H n ( G, − ) is nitary for al l n . It seems natural, therefore, to onsider groups whose n th ohomology funtors are nitary for almost al l n . W e sa y that su h a group has ohomolo gy almost everywher e nitary . In this pap er, w e shall in v estigate groups with ohomology almost ev erywhere nitary . W e b egin with the lass of lo ally (p olyyli-b y- nite) groups, and in 3 w e pro v e the follo wing: 2000 Mathematis Subje t Classi ation. 20J06 20J05 18G15. Key wor ds and phr ases. ohomology of groups, nitary funtors. 1 2 MAR TIN HAMIL TON Theorem A. L et G b e a lo al ly (p olyyli-by-nite) gr oup. Then G has ohomolo gy almost everywher e nitary if and only if G has nite virtual ohomolo gi al dimension and the normalizer of every non-trivial nite sub gr oup of G is nitely gener ate d. If G is a lo ally (p olyyli-b y-nite) group with ohomology almost ev erywhere nitary , then a result of Kropholler (Theorem 2 . 1 in [10 ℄) sho ws that G has a nite dimensional mo del for the lassifying spae E G for prop er ations and, furthermore, that there is a b ound on the orders of the nite subgroups of G (see 5 of [13℄ for a brief explanation of the lassifying spae E G ). In 9 w e pro v e the follo wing Lemma: Lemma 1.2. L et G b e a lo al ly (p olyyli-by-nite) gr oup. Then the fol lowing ar e e quivalent: (i) Ther e is a nite dimensional mo del for E G , and ther e is a b ound on the or ders of the nite sub gr oups of G ; (ii) G has nite virtual ohomolo gi al dimension; and (iii) Ther e is a nite dimensional mo del for E G , and G has nitely many onjugay lasses of nite sub gr oups. Therefore, w e an redue our study to those groups G whi h ha v e nite virtual ohomologial dimension. W e then ha v e the follo wing short exat sequene: N G ։ Q, where N is a torsion-free, lo ally (p olyyli-b y-nite) group of nite ohomologial dimension, and Q is a nite group. Hene, in order to pro v e Theorem A, w e m ust onsider three ases. The rst ase is when G is torsion-free. In this ase, G has nite ohomologial dimension, so H n ( G, − ) = 0 , and hene is nitary , for all suien tly large n . The next simplest ase, when G is the diret pro dut N × Q is treated in 2, and the general ase is then pro v ed in 3. No w, if G is an y group with ohomology almost ev erywhere nitary , and H is a subgroup of G of nite index, then it is alw a ys true that H also has ohomology almost ev erywhere nitary (see Lemma 2.1 b elo w). Ho w ev er, in the ase of lo ally (p olyyli-b y-nite) groups w e an sa y m u h more than this: Corollary B. L et G b e a lo al ly (p olyyli-by-nite) gr oup. If G has ohomolo gy almost everywher e nitary, then every sub gr oup of G also has ohomolo gy almost everywher e nitary. This is not true in general, ho w ev er, as an b een seen from Prop osi- tion 4.1 b elo w. Next, w e onsider the lass of elemen tary amenable groups, and pro v e the follo wing in 5: WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 3 Prop osition C. L et G b e an elementary amenable gr oup with ohomol- o gy almost everywher e nitary. Then G has nitely many onjugay lasses of nite sub gr oups, and C G ( E ) is nitely gener ate d for every E ≤ G of or der p . In 6 w e in v estigate the lass of groups of nite virtual ohomologial dimension. In this setion w e w ork o v er a ring R of prime harateristi p , instead of o v er Z , b y dening the n th ohomology of a group G as H n ( G, − ) := Ext n RG ( R, − ) . In order to mak e it lear that w e are no w w orking o v er R , w e sa y that H n ( G, − ) is nitary over R if and only if the funtor Ext n RG ( R, − ) is nitary . W e ha v e an analogue of Prop osition 1.1, haraterising the groups of t yp e FP ∞ o v er R as those with n th ohomology funtors nitary o v er R for all n . W e an similarly dene the notion of a group ha ving ohomolo gy almost everywher e nitary over R , and w e pro v e the follo wing result: Theorem D. L et G b e a gr oup of nite virtual ohomolo gi al dimen- sion, and R b e a ring of prime har ateristi p . Then the fol lowing ar e e quivalent: (i) G has ohomolo gy almost everywher e nitary over R ; (ii) G has nitely many onjugay lasses of elementary ab elian p - sub gr oups and the normalizer of every non-trivial elementary ab elian p -sub gr oup of G is of typ e FP ∞ over R ; and (iii) G has nitely many onjugay lasses of elementary ab elian p - sub gr oups and the normalizer of every non-trivial elementary ab elian p -sub gr oup of G has ohomolo gy almost everywher e ni- tary over R . W e then adapt the pro of of Theorem D sligh tly , and in 7 w e use it to pro v e the follo wing result, whi h answ ers a question of Leary and Nuinkis (Question 1 in [13℄): Theorem E. L et G b e a gr oup of typ e VFP over F p , and P b e a p - sub gr oup of G . Then the entr alizer C G ( P ) of P is also of typ e VFP over F p . Finally , in 8 w e return to w orking o v er Z , and onsider the lass of groups whi h p ossess a nite dimensional mo del for E G . W e pro v e the follo wing: Prop osition F. L et G b e a gr oup whih p ossesses a nite dimensional mo del for the lassifying sp a e E G for pr op er ations. If (i) G has nitely many onjugay lasses of nite sub gr oups; and 4 MAR TIN HAMIL TON (ii) The normalizer of every non-trivial nite sub gr oup of G has ohomolo gy almost everywher e nitary, Then G has ohomolo gy almost everywher e nitary. Ho w ev er, the on v erse of this result is false, and w e shall exhibit oun ter-examples in 8 b y using a theorem of Leary (Theorem 20 in [12 ℄). These oun ter-examples sho w that the on v erse of Prop osition F is false ev en for the sub lass of groups of nite virtual ohomologial dimension. 1.1. A kno wledgemen ts. I w ould lik e to thank m y resear h sup ervi- sor P eter Kropholler for all of his advie and supp ort throughout this pro jet. I w ould also lik e to thank Ian Leary for useful disussions onerning the on v erse of Prop osition F, and for suggesting that m y results ould b e used to pro v e Theorem E. 2. The Diret Pr odut Case of Theorem A Supp ose that G = N × Q , where N is a torsion-free, lo ally (p olyyli- b y-nite) group of nite ohomologial dimension, and Q is a non- trivial nite group. W e wish to sho w that G has ohomology almost ev erywhere nitary if and only if the normalizer of ev ery non-trivial - nite subgroup of G is nitely generated. No w, if F is a non-trivial nite subgroup of G , then F m ust b e a subgroup of Q , and so N is a sub- group of N G ( F ) of nite index. Hene, N G ( F ) is nitely generated if and only if N is. It is therefore enough to pro v e that G has ohomology almost ev erywhere nitary if and only if N is nitely generated. W e b egin b y assuming that N is nitely generated. Therefore N is p olyyli-b y-nite, and hene of t yp e FP ∞ (Examples 2 . 6 in [ 2℄). The prop ert y of t yp e FP ∞ is inherited b y sup ergroups of nite index, so G is also of t yp e FP ∞ . Therefore, b y Prop osition 1.1, w e see that G has ohomology almost ev erywhere nitary . F or the on v erse, w e shall pro v e a more general result whi h do es not plae an y restritions on the group N . Firstly , w e need the follo wing three lemmas: Lemma 2.1. L et G b e a gr oup, and H b e a sub gr oup of nite index. If H n ( G, − ) is nitary, then H n ( H , − ) is also nitary. Pr o of. Supp ose that H n ( G, − ) is nitary . F rom Shapiro's Lemma (Prop o- sition 6 . 2 I I I in [5℄), w e ha v e: H n ( H , − ) ∼ = H n ( G, Coind G H − ) . WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 5 Then, as H has nite index in G , it follo ws from Lemma 6 . 3 . 4 in [ 19 ℄ that Coind G H ( − ) ∼ = Ind G H ( − ) . Therefore, H n ( H , − ) ∼ = H n ( G, Ind G H − ) ∼ = H n ( G, − ⊗ Z H Z G ) , and as tensor pro duts omm ute with ltered olimits, w e see that H n ( H , − ) is the omp osite of t w o nitary funtors, and hene is itself nitary . Lemma 2.2. L et G b e a gr oup, and R 1 → R 2 b e a ring homomorphism. If H n ( G, − ) is nitary over R 1 , then H n ( G, − ) is nitary over R 2 . Pr o of. W e see from Chapter 0 of [ 2℄ that for an y R 2 G -mo dule M w e ha v e the follo wing isomorphism: Ext n R 2 G ( R 2 , M ) ∼ = Ext n R 1 G ( R 1 , M ) , where M is view ed as an R 1 G -mo dule via the homomorphism R 1 → R 2 . The result no w follo ws. Lemma 2.3. L et F 1 , F 2 : Mod R → Mod S , and supp ose that F is the dir e t sum of F 1 and F 2 . If F is nitary, then so ar e F 1 and F 2 . Pr o of. As F is the diret sum of F 1 and F 2 , w e ha v e the follo wing exat sequene of funtors: 0 → F 1 → F → F 2 → 0 . Let ( M λ ) b e a ltered olimit system of R -mo dules. W e ha v e the follo wing omm utativ e diagram with exat ro ws: lim − → λ F 1 ( M λ ) / / / / f 1 lim − → λ F ( M λ ) / / / / f lim − → λ F 2 ( M λ ) f 2 F 1 (lim − → λ M λ ) / / / / F (lim − → λ M λ ) / / / / F 2 (lim − → λ M λ ) As F is nitary , w e see that the map f is an isomorphism. It then follo ws from the Snak e Lemma that f 1 is a monomorphism and f 2 is an epimorphism. No w, as F is the diret sum of F 1 and F 2 , w e also ha v e the follo wing exat sequene of funtors: 0 → F 2 → F → F 1 → 0 , and hene the follo wing omm utativ e diagram with exat ro ws: lim − → λ F 2 ( M λ ) / / / / f 2 lim − → λ F ( M λ ) / / / / f lim − → λ F 1 ( M λ ) f 1 F 2 (lim − → λ M λ ) / / / / F (lim − → λ M λ ) / / / / F 1 (lim − → λ M λ ) 6 MAR TIN HAMIL TON and a similar argumen t to ab o v e sho ws that f 2 is a monomorphism and f 1 is an epimorphism. The result no w follo ws. Prop osition 2.4. L et Q b e a non-trivial nite gr oup, and N b e any gr oup. If ther e is some natur al numb er k suh that H k ( N × Q, − ) is nitary, then N is nitely gener ate d. Pr o of. Supp ose that H k ( N × Q, − ) is nitary . As Q is a non-trivial nite group, w e an ho ose a subgroup E of Q of order p , for some prime p , so N × E is a subgroup of N × Q of nite index. It then follo ws from Lemma 2.1 that H k ( N × E , − ) is also nitary . Then, b y Lemma 2.2 , w e see that H k ( N × E , − ) is nitary o v er F p . Let M b e an y F p N -mo dule, and F p b e the trivial F p E -mo dule. Ap- plying the Künneth Theorem giv es the follo wing isomorphism: H k ( N × E , M ) ∼ = L i + j = k H i ( N , M ) ⊗ F p H j ( E , F p ) ∼ = L k i =0 H i ( N , M ) , and as this holds for an y F p N -mo dule M , w e ha v e an isomorphism of funtors for mo dules on whi h E ats trivially . Then, as H k ( N × E , − ) is nitary o v er F p , it follo ws from Lemma 2.3 that H 0 ( N , − ) is also nitary o v er F p . It then follo ws that N is nitely generated (see, for example, Prop osition 2 . 1 in [2℄). The on v erse of the diret pro dut ase no w follo ws immediately . 3. Pr oof of Theorem A Let G b e a lo ally (p olyyli-b y-nite) group of nite virtual o- homologial dimension. W e b egin with the follo wing useful result of Corni k and Kropholler (Theorem A in [ 7℄): Prop osition 3.1. L et G b e a gr oup p ossessing a nite dimensional mo del for E G , and M b e an RG -mo dule. Then M has nite pr oje tive dimension over RG if and only if M has nite pr oje tive dimension over RH for al l nite sub gr oups H of G . Theorem 3.2. L et G b e a lo al ly (p olyyli-by-nite) gr oup of nite virtual ohomolo gi al dimension. If G has ohomolo gy almost every- wher e nitary, then the normalizer of every non-trivial nite sub gr oup of G is nitely gener ate d. Pr o of. Let F b e a non-trivial nite subgroup of G , so w e an ho ose a subgroup E of F of order p , for some prime p . As G has nite virtual ohomologial dimension, it has a torsion-free normal subgroup N of WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 7 nite index. Let H := N E , so it follo ws from Lemma 2.1 that H has ohomology almost ev erywhere nitary . Let Λ denote the set of non-trivial nite subgroups of H , so Λ onsists of subgroups of order p . No w H ats on this set b y onjugation, so the stabilizer of an y K ∈ Λ is N H ( K ) . Also, for ea h K ∈ Λ , w e see that the set of K -xed p oin ts Λ K is simply the set { K } , b eause if K xed some K ′ 6 = K , then K K ′ w ould b e a subgroup of H of order p 2 , whi h is a on tradition. W e ha v e the follo wing short exat sequene: J Z Λ ε ։ Z , where ε denotes the augmen tation map. F or ea h K ∈ Λ , w e see that J is free as a Z K -mo dule with basis { K ′ − K : K ′ ∈ Λ } . No w, as H has nite virtual ohomologial dimension, it has a nite dimensional mo del for E H (Exerise VI I I. 3 in [5℄), and so it follo ws from Prop osition 3.1 that J has nite pro jetiv e dimension o v er Z H . No w, the short exat sequene J Z Λ ։ Z giv es rise to a long exat sequene in ohomology , and as J has nite pro jetiv e dimension, w e onlude that for all suien tly large n w e ha v e the follo wing isomorphism: H n ( H , − ) ∼ = Ext n Z H ( Z Λ , − ) . Next, as H ats on Λ , w e an split Λ up in to its H -orbits, so Λ = a K ∈ C H K \ H = a K ∈ C N H ( K ) \ H , where K runs through a set C of represen tativ es of onjugay lasses of non-trivial nite subgroups of H . This giv es the follo wing isomorphism: H n ( H , − ) ∼ = Q K ∈ C Ext n Z H ( Z [ N H ( K ) \ H ] , − ) ∼ = Q K ∈ C H n ( N H ( K ) , − ) , where the last isomorphism follo ws from the E kmannShapiro Lemma. Therefore, if H n ( H , − ) is nitary , it follo ws from Lemma 2.3 that H n ( N H ( E ) , − ) is also nitary . Hene, as H has ohomology almost ev erywhere nitary , w e onlude that N H ( E ) also has ohomology al- most ev erywhere nitary . No w, as E is a nite group, | N H ( E ) : C H ( E ) | < ∞ , and so b y Lemma 2.1 w e see that C H ( E ) ∼ = E × C N ( E ) 8 MAR TIN HAMIL TON has ohomology almost ev erywhere nitary . It then follo ws from Prop o- sition 2.4 that C N ( E ) is nitely generated, and hene p olyyli-b y- nite. No w, as E ≤ F , it follo ws that C N ( F ) ≤ C N ( E ) and as ev ery subgroup of a p olyyli-b y-nite group is nitely generated, w e see that C N ( F ) is nitely generated. Finally , as N is a subgroup of G of nite index, it follo ws that | C G ( F ) : C N ( F ) | < ∞ , and so C G ( F ) is nitely generated. Hene N G ( F ) is nitely generated, as required. In the remainder of this setion w e shall pro v e the on v erse. Firstly , w e need the follo wing denition from [11℄: Denition 3.3. Let G b e a group, and let Λ( G ) denote the p oset of the non-trivial nite subgroups of G . W e an view this p oset as a G -simpliial omplex, whi h w e shall denote b y | Λ( G ) | , b y the fol- lo wing metho d: An n -simplex in | Λ( G ) | is determined b y ea h stritly inreasing hain H 0 < H 1 < · · · < H n of n + 1 non-trivial nite subgroups of G . The ation of G on the set of non-trivial nite subgroups indues an ation of G on | Λ( G ) | , so that the stabilizer of a simplex is an in tersetion of normalizers; in the ase of the simplex determined b y the hain of subgroups ab o v e, the stabilizer is n \ i =0 N G ( H i ) . This omplex has the prop ert y that, for an y non-trivial nite subgroup K of G , the K -xed p oin t omplex | Λ( G ) | K is on tratible (for a pro of of this, see Lemma 2 . 1 in [11 ℄). Next, w e need the follo wing t w o results of Kropholler and Mislin: Prop osition 3.4. L et Y b e a G -CW- omplex of nite dimension n . Then Y an b e emb e dde d into an n -dimensional G -CW- omplex e Y whih is ( n − 1) - onne te d in suh a way that G ats fr e ely outside Y . Pr o of. This is Lemma 4 . 4 of [11℄. W e an tak e e Y to b e the n -sk eleton of the join Y ∗ G ∗ · · · ∗ G | {z } n . WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 9 Prop osition 3.5. L et Y b e an n -dimensional G -CW- omplex whih is ( n − 1) - onne te d, for some n ≥ 0 . Supp ose that Y K is ontr atible for al l non-trivial nite sub gr oups K of G . Then the n th r e du e d homolo gy gr oup f H n ( Y ) is pr oje tive as a Z K -mo dule for al l nite sub gr oups K of G . Pr o of. This is Prop osition 6 . 2 of [ 11℄. Finally , w e require the follo wing t w o lemmas: Lemma 3.6. L et 0 → F 1 → F → F 2 → 0 b e an exat se quen e of funtors fr om Mod R to Mod S . If F 1 and F 2 ar e nitary, then so is F . Pr o of. Let ( M λ ) b e a ltered olimit system of R -mo dules. W e ha v e the follo wing omm utativ e diagram: 0 / / lim − → λ F 1 ( M λ ) / / f 1 lim − → λ F ( M λ ) / / f lim − → λ F 2 ( M λ ) / / f 2 0 0 / / F 1 (lim − → λ M λ ) / / F (lim − → λ M λ ) / / F 2 (lim − → λ M λ ) / / 0 No w, as F 1 and F 2 are nitary , the maps f 1 and f 2 are isomorphisms. It then follo ws from the Fiv e Lemma that f is an isomorphism, and w e onlude that F is nitary . Lemma 3.7. L et G b e a gr oup. If we have an exat se quen e of Z G - mo dules 0 → A r → A r − 1 → · · · → A 0 → Z → 0 suh that, for e ah i = 0 , . . . , r , the funtor Ext ∗ Z G ( A i , − ) is nitary in al l suiently high dimensions, then G has ohomolo gy almost every- wher e nitary. Pr o of. If r = 0 , then the result follo ws immediately . Assume, therefore, that r ≥ 1 , and pro eed b y indution. If r = 1 , then w e ha v e the short exat sequene A 1 A 0 ։ Z whi h giv es the follo wing long exat sequene: · · · → Ext j Z G ( A 0 , − ) → Ext j Z G ( A 1 , − ) → H j +1 ( G, − ) → → Ext j +1 Z G ( A 0 , − ) → Ext j +1 Z G ( A 1 , − ) → · · · 10 MAR TIN HAMIL TON and as b oth Ext ∗ Z G ( A 0 , − ) and Ext ∗ Z G ( A 1 , − ) are nitary in all su- ien tly high dimensions, it follo ws from the Fiv e Lemma that G has ohomology almost ev erywhere nitary . No w supp ose that w e ha v e sho wn this for r − 1 , and that w e ha v e an exat sequene 0 → A r → A r − 1 → · · · → A 0 → Z → 0 su h that, for ea h i = 0 , . . . , r , the funtor Ext ∗ Z G ( A i , − ) is nitary in all suien tly high dimensions. Let K := Ker( A r − 2 → A r − 3 ) , so w e ha v e the short exat sequene A r A r − 1 ։ K , and an argumen t similar to ab o v e sho ws that Ext ∗ Z G ( K , − ) is nitary in all suien tly high dimensions. W e then ha v e the follo wing exat sequene: 0 → K → A r − 2 → · · · → A 0 → Z → 0 , and the result no w follo ws b y indution. Finally , w e an no w pro v e the on v erse: Theorem 3.8. L et G b e a lo al ly (p olyyli-by-nite) gr oup of nite virtual ohomolo gi al dimension. If the normalizer of every non-trivial nite sub gr oup of G is nitely gener ate d, then G has ohomolo gy almost everywher e nitary. Pr o of. Let Λ( G ) b e the p oset of all non-trivial nite subgroups of G , and let | Λ( G ) | denote its realization as a G -simpliial omplex. As G has nite virtual ohomologial dimension, there is a b ound on the orders of its nite subgroups, and so | Λ( G ) | is nite-dimensional, sa y dim | Λ( G ) | = r . F rom Prop osition 3.4 , w e an em b ed | Λ( G ) | in to an r -dimensional G -CW-omplex Y whi h is ( r − 1) -onneted, su h that G ats freely outside | Λ( G ) | . Consider the augmen ted ellular hain omplex of Y . As Y is ( r − 1) -onneted, it has trivial homology exept in dimension r , and so w e ha v e the follo wing exat sequene: 0 → e H r ( Y ) → C r ( Y ) → · · · → C 0 ( Y ) → Z → 0 . In order to sho w that G has ohomology almost ev erywhere nitary , it is enough b y Lemma 3.7 to sho w that the funtors Ext ∗ Z G ( e H r ( Y ) , − ) and Ext ∗ Z G ( C l ( Y ) , − ) , 0 ≤ l ≤ r , are nitary in all suien tly high dimensions. Firstly , notie that for ev ery non-trivial nite subgroup K of G , Y K = | Λ( G ) | K , as the opies of G that w e ha v e added in the onstru- tion of Y ha v e free orbits, and so ha v e no xed p oin ts under K . Th us, WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 11 Y is an r -dimensional G -CW-omplex whi h is ( r − 1) -onneted, su h that Y K is on tratible for ev ery non-trivial nite subgroup K of G . It then follo ws from Prop osition 3.5 that e H r ( Y ) is pro jetiv e as a Z K - mo dule for all nite subgroups K of G . Then b y Prop osition 3.1 , e H r ( Y ) has nite pro jetiv e dimension o v er Z G , and so Ext n Z G ( e H r ( Y ) , − ) = 0 , and th us is nitary , for all suien tly large n . Next, for ea h 0 ≤ l ≤ r , onsider the funtor Ext ∗ Z G ( C l ( Y ) , − ) . Pro vided that n ≥ 1 , w e see that Ext n Z G ( C l ( Y ) , − ) ∼ = Ext n Z G ( C l ( | Λ( G ) | ) , − ) as the opies of G that w e ha v e added in the onstrution of Y ha v e free orbits, and so the free-ab elian group on them is a free mo dule. No w, Ext n Z G ( C l ( | Λ( G ) | ) , − ) ∼ = Ext n Z G ( Z | Λ( G ) | l , − ) , where | Λ( G ) | l onsists of all the l -simpliies K 0 < K 1 < · · · < K l in | Λ( G ) | . As G ats on | Λ( G ) | l , w e an therefore split | Λ( G ) | l up in to its G -orbits, where the stabilizer of su h a simplex is T l i =0 N G ( K i ) . W e then obtain the follo wing isomorphism: Ext n Z G ( Z | Λ( G ) | l , − ) ∼ = Ext n Z G ( Z [ ` C T l i =0 N G ( K i ) \ G ] , − ) ∼ = Q C Ext n Z G ( Z [ T l i =0 N G ( K i ) \ G ] , − ) ∼ = Q C H n ( T l i =0 N G ( K i ) , − ) , where the pro dut is tak en o v er a set C of represen tativ es of onjugay lasses of non-trivial nite subgroups of G . No w, as G has nite virtual ohomologial dimension, it follo ws from Lemma 1.2 that there are only nitely man y onjugay lasses of nite subgroups, and so this pro dut is nite. No w, for ea h l -simplex K 0 < · · · < K l w e ha v e l \ i =0 N G ( K i ) ≤ N G ( K l ) . Then, as N G ( K l ) is nitely generated, it follo ws that T l i =0 N G ( K i ) is also nitely generated, and hene p olyyli-b y-nite. Therefore, T l i =0 N G ( K i ) is of t yp e FP ∞ , and so b y Prop osition 1.1 , H n ( T l i =0 N G ( K i ) , − ) is nitary . Th us Ext n Z G ( C l ( Y ) , − ) is isomorphi to a nite pro dut of nitary funtors, and hene b y Lemma 3.6 is nitary . As this holds for all n ≥ 1 , w e see that Ext ∗ Z G ( C l ( Y ) , − ) is nitary in all suien tly high dimensions, whi h ompletes the pro of. 12 MAR TIN HAMIL TON 4. Pr oof of Cor ollar y B Corollary B. L et G b e a lo al ly (p olyyli-by-nite) gr oup. If G has ohomolo gy almost everywher e nitary, then every sub gr oup of G also has ohomolo gy almost everywher e nitary. Pr o of. As G has ohomology almost ev erywhere nitary , it follo ws from Theorem A that G has nite virtual ohomologial dimension and the normalizer of ev ery non-trivial nite subgroup of G is nitely generated. Let H b e an y subgroup of G , so v cd H ≤ vcd G < ∞ . Also, let F b e a non-trivial nite subgroup of H . Then N G ( F ) is nitely generated, hene p olyyli-b y-nite, and as N H ( F ) ≤ N G ( F ) , w e see that N H ( F ) is also nitely generated. Therefore, w e onlude from Theorem A that H has ohomology almost ev erywhere nitary . This result do es not hold in general, ho w ev er, as the follo wing prop o- sition sho ws: Prop osition 4.1. L et G b e a gr oup of typ e FP ∞ whih has an innitely gener ate d sub gr oup H , and let Q b e a non-trivial nite gr oup. Then G × Q has ohomolo gy almost everywher e nitary, but H × Q do es not. Pr o of. As G is of t yp e FP ∞ , it follo ws that G × Q is also of t yp e FP ∞ , and so has ohomology almost ev erywhere nitary . Ho w ev er, as H is innitely generated, it follo ws from Prop osition 2.4 that H n ( H × Q, − ) is not nitary for an y n . R emark 4.2 . Let G b e the free group on t w o generators x, y , so G is of t yp e FP ∞ (Example 2.6 in [2℄), and let H b e the subgroup of G generated b y y n xy − n for all n . W e then ha v e a oun ter-example sho wing that Corollary B do es not hold in general. 5. A Resul t on Element ar y Amenable Gr oups Prop osition C. L et G b e an elementary amenable gr oup with ohomol- o gy almost everywher e nitary. Then G has nitely many onjugay lasses of nite sub gr oups, and C G ( E ) is nitely gener ate d for every E ≤ G of or der p . Pr o of. Let G b e an elemen tary amenable group with ohomology al- most ev erywhere nitary . Kropholler's Theorem (Theorem 2 . 1 in [ 10 ℄) applies to a large lass of groups, whi h inludes all elemen tary amenable WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 13 groups. This theorem implies that G has a nite dimensional mo del for E G , and that G has a b ound on the orders of its nite subgroups. The pro of of Lemma 1.2 generalizes immediately to the elemen tary amenable ase, and w e onlude that G has nitely man y onjugay lasses of nite subgroups, and furthermore that G has nite virtual o- homologial dimension. Therefore, w e an ho ose a torsion-free normal subgroup N of G of nite index. Let E b e an y subgroup of G of order p , and let H := N E . F ollo wing the pro of of Theorem 3.2, w e see that N H ( E ) has ohomology almost ev erywhere nitary . Hene, C H ( E ) ∼ = E × C N ( E ) also has ohomology almost ev erywhere nitary , and so b y Prop osition 2.4 w e see that C N ( E ) is nitely generated. The result no w follo ws. 6. Generaliza tion to Gr oups of Finite Vir tual Cohomologial Dimension In this setion, w e shall pro v e Theorem D. It sues to sho w this for the ase R = F p , b y the follo wing lemma: Lemma 6.1. L et G b e a gr oup, and R b e a ring of prime har ateristi p . Then H n ( G, − ) is nitary over R if and only if H n ( G, − ) is nitary over F p . Pr o of. If H n ( G, − ) is nitary o v er F p , then it follo ws from Lemma 2.2 that H n ( G, − ) is nitary o v er R . Con v ersely , supp ose that H n ( G, − ) is nitary o v er R ; that is, the funtor Ext n RG ( R, − ) is nitary . Let ( M λ ) b e a ltered olimit system of F p G -mo dules. Then ( M λ ⊗ F p R ) is a ltered olimit system of RG - mo dules, and so the natural map lim − → λ Ext n RG ( R, M λ ⊗ F p R ) → Ext n RG ( R, lim − → λ M λ ⊗ F p R ) is an isomorphism. No w, as an F p -v etor spae, R ∼ = F p ⊕ V for some F p -v etor spae V . Therefore, for ea h λ , M λ ⊗ F p R ∼ = M λ ⊕ M λ ⊗ F p V , and so Ext n RG ( R, M λ ⊗ F p R ) ∼ = Ext n RG ( R, M λ ) ⊕ Ext n RG ( R, M λ ⊗ F p V ) . It then follo ws from Lemma 2.3 that the natural map lim − → λ Ext n RG ( R, M λ ) → Ext n RG ( R, lim − → λ M λ ) 14 MAR TIN HAMIL TON is an isomorphism. No w, w e see from Chapter 0 of [2℄ that Ext n RG ( R, − ) ∼ = Ext n F p G ( F p , − ) on F p G -mo dules, so therefore it follo ws that the natural map lim − → λ Ext n F p G ( F p , M λ ) → Ext n F p G ( F p , lim − → λ M λ ) is an isomorphism, and hene that H n ( G, − ) is nitary o v er F p . 6.1. Pro of of (i) ⇒ (ii). Let G b e a group of nite virtual ohomologial dimension with ohomology almost ev erywhere nitary o v er F p . W e b egin b y sho wing that the normalizer of ev ery non-trivial elemen tary ab elian p -subgroup of G is of t yp e FP ∞ o v er F p . In fat, w e shall sho w that the normalizer of ev ery non-trivial nite p -subgroup of G is of t yp e FP ∞ o v er F p . Lemma 6.2. L et N b e any gr oup, and Q b e a non-trivial nite gr oup whose or der is divisible by p . If N × Q has ohomolo gy almost every- wher e nitary over F p , then N × Q is of typ e FP ∞ over F p . Pr o of. Supp ose that N × Q is not of t yp e FP ∞ o v er F p , so N is not of t yp e FP ∞ o v er F p . Therefore, there is some n su h that H n ( N , − ) is not nitary o v er F p . Let E b e a subgroup of Q of order p , so b y an argumen t similar to the pro of of Prop osition 2.4 w e obtain, for ea h m , the follo wing isomorphism of funtors: H m ( N × E , − ) ∼ = m M i =0 H i ( N , − ) , for mo dules on whi h E ats trivially . As H n ( N , − ) is not nitary o v er F p , it follo ws from Lemma 2.3 that H m ( N × E , − ) is not nitary o v er F p for all m ≥ n . Therefore, b y an easy generalization of Lemma 2.1, w e see that H m ( N × Q, − ) is not nitary o v er F p for all m ≥ n , whi h is a on tradition. Lemma 6.3. L et G b e a gr oup of nite virtual ohomolo gi al dimension with ohomolo gy almost everywher e nitary over F p , and let E b e a sub gr oup of or der p . Then the normalizer N G ( E ) of E is of typ e FP ∞ over F p . Pr o of. As G has nite virtual ohomologial dimension, w e an ho ose a torsion-free normal subgroup N of nite index. Let H := N E . A sligh t v ariation on the pro of of Theorem 3.2 sho ws that N H ( E ) has ohomology almost ev erywhere nitary o v er F p . Therefore, C H ( E ) ∼ = E × C N ( E ) WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 15 has ohomology almost ev erywhere nitary o v er F p , and b y Lemma 6.2 , w e see that C H ( E ) is of t yp e FP ∞ o v er F p . Th us, N G ( E ) is of t yp e FP ∞ o v er F p , as required. Theorem 6.4. L et G b e a gr oup of nite virtual ohomolo gi al dimen- sion with ohomolo gy almost everywher e nitary over F p , and let F b e a non-trivial nite p -sub gr oup. Then the normalizer N G ( F ) of F is of typ e FP ∞ over F p . Pr o of. Supp ose that F has order p k , where k ≥ 1 . W e pro eed b y indution on k . If k = 1 , then the result follo ws from Lemma 6.3 . Supp ose no w that k ≥ 2 . As the en tre ζ ( F ) of F is non-trivial, w e an ho ose a subgroup E ≤ ζ ( F ) of order p . Then C G ( E ) is of t yp e FP ∞ o v er F p b y Lemma 6.3 , and Prop osition 2 . 7 in [2℄ sho ws that C G ( E ) / E is also of t yp e FP ∞ o v er F p . By indution, the normalizer of F /E in C G ( E ) / E , whi h is ( N G ( F ) ∩ C G ( E ) ) /E , is of t yp e FP ∞ o v er F p . Another appliation of Prop osition 2 . 7 in [2℄ sho ws that N G ( F ) ∩ C G ( E ) is of t yp e FP ∞ o v er F p , and as C G ( F ) ≤ N G ( F ) ∩ C G ( E ) ≤ N G ( F ) , w e see that N G ( F ) is of t yp e FP ∞ o v er F p . Next, w e shall sho w that G has nitely man y onjugay lasses of elemen tary ab elian p -subgroups. Firstly , w e need the follo wing lemma: Lemma 6.5. L et G b e a gr oup. If H n ( G, − ) is nitary over F p , then H n ( G, F p ) is nite-dimensional as an F p -ve tor sp a e. Pr o of. Supp ose that H n ( G, F p ) is innite-dimensional as an F p -v etor spae. By the Univ ersal Co eien t Theorem, w e ha v e the follo wing isomorphism: H n ( G, F p ) ∼ = Hom F p ( H n ( G, F p ) , F p ) . Hene H n ( G, F p ) is also innite-dimensional as an F p -v etor spae, with basis { e i : i ∈ I } , sa y . W e then ha v e: H n ( G, F p ) ∼ = Y I F p . 16 MAR TIN HAMIL TON Next, let L J F p b e an innite diret sum of opies of F p . As H n ( G, − ) is nitary o v er F p , the natural map M J H n ( G, F p ) → H n ( G, M J F p ) is an isomorphism; that is, M J Y I F p ∼ = Y I M J F p , whi h is learly a on tradition. Next, reall the follo wing denition from [8℄: Denition 6.6. A homomorphism φ : A → B of F p -algebras is alled a uniform F -isomorphism if and only if there exists a natural n um b er n su h that: • If x ∈ Ker φ , then x p n = 0 ; and • If y ∈ B , then y p n is in the image of φ . W e also ha v e the follo wing result of Henn (Theorem A. 4 in [ 8℄): Prop osition 6.7. If G is a disr ete gr oup suh that ther e exists a nite- dimensional ontr atible G -CW- omplex X with al l el l stabilizers nite of b ounde d or der, then ther e exists a uniform F -isomorphism φ : H ∗ ( G, F p ) → lim A p ( G ) op H ∗ ( E , F p ) , wher e A p ( G ) denotes the ate gory with obje ts the elementary ab elian p -sub gr oups E of G , and morphisms the gr oup homomorphisms whih an b e indu e d by onjugation by an element of G . Finally , w e an pro v e the follo wing prop osition, whi h is a general- ization of a result of Henn (Theorem A. 8 in [ 8℄): Prop osition 6.8. L et G b e a gr oup of nite virtual ohomolo gi al di- mension with ohomolo gy almost everywher e nitary over F p . Then G has nitely many onjugay lasses of elementary ab elian p -sub gr oups. Pr o of. As G has nite virtual ohomologial dimension, there is a - nite dimensional mo del, sa y X , for the lassifying spae E G for prop er ations (Exerise VI I I. 3 in [5℄). Th us, X is a nite dimensional on- tratible G -CW-omplex with all ell stabilizers nite of b ounded order, so it follo ws from Prop osition 6.7 that there is a uniform F -isomorphism φ : H ∗ ( G, F p ) → lim A p ( G ) op H ∗ ( E , F p ) . No w assume that there are innitely man y onjugay lasses of elemen- tary ab elian p -subgroups of G . As the order of the nite subgroups is WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 17 b ounded, this means that there m ust b e innitely man y maximal el- emen tary ab elian p -subgroups of G of the same rank k (although k itself need not neessarily b e maximal). F ollo wing Henn's argumen t, w e an use this fat to onstrut innitely man y linearly indep enden t non-nilp oten t lasses in the in v erse limit in some degree (for the de- tails, see the pro of of Theorem A. 8 in [8℄). No w, raising these to a large enough p o w er and using the fat that φ is a uniform F -isomorphism, w e see that H ∗ ( G, F p ) is innite-dimensional as an F p -v etor spae in some degree m su h that H m ( G, − ) is nitary o v er F p . This giv es a on tradition to Lemma 6.5. 6.2. Pro of of (ii) ⇒ (iii). This is immediate. 6.3. Pro of of (iii) ⇒ (i). Let G b e a group of nite virtual ohomologial dimension, su h that G has nitely man y onjugay lasses of elemen tary ab elian p - subgroups and the normalizer of ev ery non-trivial elemen tary ab elian p -subgroup of G has ohomology almost ev erywhere nitary o v er F p . W e shall sho w that G has ohomology almost ev erywhere nitary o v er F p . Firstly , let A p ( G ) denote the p oset of all the non-trivial elemen tary ab elian p -subgroups of G , and let S p ( G ) denote the p oset of all the non-trivial nite p -subgroups of G . W e see from Remark 2.3(i) in [ 17 ℄ that the inlusion of p osets A p ( G ) ֒ → S p ( G ) indues a G -homotop y equiv alene |A p ( G ) | ≃ G |S p ( G ) | b et w een the G -simpliial omplexes. Next, w e need the follo wing result (Prop osition 2 . 7 I I in [3 ℄): Prop osition 6.9. L et X and Y b e G -CW- omplexes, and φ : X → Y b e a G -e quivariant el lular map. Then φ is a G -homotopy e quivalen e if and only if φ H : X H → Y H is a homotopy e quivalen e for al l sub gr oups H of G . W e an no w pro v e the follo wing k ey lemma: Lemma 6.10. The omplex |A p ( G ) | E is ontr atible for al l E ∈ A p ( G ) . Pr o of. W e follo w an argumen t similar to the pro of of Lemma 2 . 1 in [11 ℄: If H ∈ S p ( G ) E , then E H is a p -subgroup of G . W e an therefore dene a funtion f : S p ( G ) E → S p ( G ) E 18 MAR TIN HAMIL TON b y f ( H ) = E H , so for all H ∈ S p ( G ) E w e ha v e: H ≤ f ( H ) ≥ E . W e then see that S p ( G ) E is onially on tratible in the sense of Quillen (see 1 . 5 in [15℄), whi h implies that |S p ( G ) E | is on tratible b y Quillen's argumen t. Finally , b y Prop osition 6.9 , w e see that |A p ( G ) | E ≃ |S p ( G ) | E = |S p ( G ) E | , and the result no w follo ws. Finally , w e an no w pro v e the follo wing: Theorem 6.11. L et G b e a gr oup of nite virtual ohomolo gi al dimen- sion. If G has nitely many onjugay lasses of elementary ab elian p -sub gr oups, and the normalizer of every non-trivial elementary ab elian p -sub gr oup of G has ohomolo gy almost everywher e nitary over F p , then G has ohomolo gy almost everywher e nitary over F p . Pr o of. Let A p ( G ) b e the p oset of all non-trivial elemen tary ab elian p - subgroups of G , and let |A p ( G ) | denote its realization as a G -simpliial omplex. As G has nitely man y onjugay lasses of elemen tary ab elian p -subgroups, there m ust b e a b ound on their orders, and so |A p ( G ) | is nite-dimensional, sa y dim |A p ( G ) | = r . By Prop osition 3.4 , w e an em b ed |A p ( G ) | in to an r -dimensional G -CW-omplex Y whi h is ( r − 1) -onneted, su h that G ats freely outside |A p ( G ) | . The augmen ted ellular hain omplex of Y then giv es the follo wing exat sequene of Z G -mo dules: 0 → e H r ( Y ) → C r ( Y ) → · · · → C 0 ( Y ) → Z → 0 , whi h giv es the follo wing exat sequene of F p G -mo dules: 0 → e H r ( Y ) ⊗ F p → C r ( Y ) ⊗ F p → · · · → C 0 ( Y ) ⊗ F p → F p → 0 . In order to sho w that G has ohomology almost ev erywhere nitary o v er F p , it is enough b y an easy generalization of Lemma 3.7 to sho w that the funtors Ext ∗ F p G ( e H r ( Y ) ⊗ F p , − ) and Ext ∗ F p G ( C l ( Y ) ⊗ F p , − ) , 0 ≤ l ≤ r , are nitary in all suien tly high dimensions. Firstly , notie that for ev ery E ∈ A p ( G ) , Y E = |A p ( G ) | E , and hene is on tratible, as the opies of G w e ha v e added in the onstrution of Y ha v e free orbits, and so ha v e no xed p oin ts under E . Therefore, an easy generalization of Prop osition 3.5 sho ws that e H r ( Y ) ⊗ F p is pro jetiv e as an F p E -mo dule for all elemen tary ab elian p -subgroups E of G . WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 19 Let K b e a nite subgroup of G , so e H r ( Y ) ⊗ F p restrited to K is an F p K -mo dule with the prop ert y that its restrition to ev ery elemen tary ab elian p -subgroup of K is pro jetiv e. It then follo ws from Chouinard's Theorem [6 ℄ that e H r ( Y ) ⊗ F p is pro jetiv e as an F p K -mo dule. As this holds for ev ery nite subgroup K of G , it then follo ws from Prop osition 3.1 that e H r ( Y ) ⊗ F p has nite pro jetiv e dimension o v er F p G . Hene Ext n F p G ( e H r ( Y ) ⊗ F p , − ) = 0 , and th us is nitary , for all suien tly large n . Next, for ea h 0 ≤ l ≤ r , onsider the funtor Ext ∗ F p G ( C l ( Y ) ⊗ F p , − ) . Pro vided that n ≥ 1 , w e see that Ext n F p G ( C l ( Y ) ⊗ F p , − ) ∼ = Ext n F p G ( C l ( |A p ( G ) | ) ⊗ F p , − ) ∼ = Ext n F p G ( F p |A p ( G ) | l , − ) , where |A p ( G ) | l onsists of all the l -simpliies E 0 < E 1 < · · · < E l in |A p ( G ) | . As G ats on |A p ( G ) | l , w e an therefore split |A p ( G ) | l up in to its G -orbits, where the stabilizer of su h a simplex is T l i =0 N G ( E i ) . W e then obtain the follo wing isomorphism: Ext n F p G ( F p |A p ( G ) | l , − ) ∼ = Ext n F p G ( F p [ ` C T l i =0 N G ( E i ) \ G ] , − ) ∼ = Q C Ext n F p G ( F p [ T l i =0 N G ( E i ) \ G ] , − ) ∼ = Q C H n ( T l i =0 N G ( E i ) , − ) , where the pro dut is tak en o v er a set C of represen tativ es of onjugay lasses of non-trivial elemen tary ab elian p -subgroups of G . As w e are assuming that G has only nitely man y su h onjugay lasses, this pro dut is nite. No w, for ea h l -simplex E 0 < E 1 < · · · < E l w e ha v e C G ( E l ) ≤ l \ i =0 N G ( E i ) ≤ N G ( E l ) , and so | N G ( E l ) : l \ i =0 N G ( E i ) | < ∞ . Then, as N G ( E l ) has ohomology almost ev erywhere nitary o v er F p , w e see from an easy generalization of Lemma 2.1 that T l i =0 N G ( E i ) has ohomology almost ev erywhere nitary o v er F p , and so for all su- ien tly large n , H n ( T l i =0 N G ( E i ) , − ) is nitary o v er F p . Therefore, for all suien tly large n , Ext n F p G ( C l ( Y ) ⊗ F p , − ) is isomorphi to a nite 20 MAR TIN HAMIL TON pro dut of nitary funtors, and hene is nitary , whi h ompletes the pro of. 7. A Question of Lear y and Nuinkis In [13℄, Leary and Nuinkis p osed the follo wing question: If G is a group of t yp e VFP o v er F p , and P is a p -subgroup of G , is the en tralizer C G ( P ) of P neessarily of t yp e VFP o v er F p ? In this setion, w e shall giv e a p ositiv e answ er to this question. Firstly , reall (see 2 of [13℄) that a group G is said to b e of t yp e VFP o v er F p if and only if it has a subgroup of nite index whi h is of t yp e FP o v er F p . Prop osition 7.1. L et G b e a gr oup whih has a sub gr oup H of nite index with cd F p H < ∞ . Then ther e exists a nite dimensional G -CW- omplex X with nite el l stabilizers suh that 0 → C r ( X ) ⊗ F p → · · · → C 0 ( X ) ⊗ F p → F p → 0 is an exat se quen e of F p G -mo dules. Pr o of. As cd F p H < ∞ , it follo ws from an easy generalization of Theo- rem 7 . 1 VI I I in [5℄ that there exists a nite dimensional free H -CW- omplex X ′ with the prop ert y that e C ∗ ( X ′ ) ⊗ F p is exat. Set X := Hom H ( G, X ′ ) . An easy generalization of the pro of of Theorem 3 . 1 VI I I in [5℄ then sho ws that X has the required prop erties. Next, w e pro v e the follo wing k ey lemma, whi h is a v ariation on Prop osition 3.1: Lemma 7.2. L et G b e a gr oup of typ e VFP over F p , and M b e an F p G - mo dule. If M is pr oje tive as an F p K -mo dule for al l nite sub gr oups K of G , then M has nite pr oje tive dimension over F p G . Pr o of. As G is of t yp e VFP o v er F p , w e see from Prop osition 7.1 that there exists a nite-dimensional G -CW-omplex X with nite ell sta- bilizers, su h that 0 → C r ( X ) ⊗ F p → · · · → C 0 ( X ) ⊗ F p → F p → 0 is exat. No w, for ea h k , C k ( X ) is a p erm utation mo dule, C k ( X ) ∼ = M σ ∈ Σ k Z [ G σ \ G ] , WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 21 where Σ k is a set of G -orbit represen tativ es of k -ells in X , and G σ is the stabilizer of σ . If w e tensor the ab o v e exat sequene with M , then w e obtain the follo wing: 0 → M ⊗ F p ( C r ( X ) ⊗ F p ) → · · · → M ⊗ F p ( C 0 ( X ) ⊗ F p ) → M → 0 , where, for ea h k , w e ha v e M ⊗ F p ( C k ( X ) ⊗ F p ) ∼ = M σ ∈ Σ k M ⊗ F p G σ F p G. No w, as M is pro jetiv e as an F p G σ -mo dule, w e see that M ⊗ F p G σ F p G is pro jetiv e as an F p G -mo dule. Therefore, M ⊗ F p ( C k ( X ) ⊗ F p ) is pro jetiv e as an F p G -mo dule, and so the ab o v e exat sequene is a pro jetiv e resolution of M , and w e then onlude that M has nite pro jetiv e dimension o v er F p G . W e an no w answ er Leary and Nuinkis' question in the ase where P has order p . This is a v ariation on Lemma 6.3: Prop osition 7.3. L et G b e a gr oup of typ e VFP over F p , and let P b e a sub gr oup of G of or der p . Then C G ( P ) is of typ e VFP over F p . Pr o of. As G is of t yp e VFP o v er F p , w e an ho ose a normal subgroup N of nite index whi h is of t yp e FP o v er F p . Let H := N P , so H is of t yp e VFP o v er F p . Next, let A p ( H ) denote the set of all non-trivial elemen tary ab elian p -subgroups of H , so A p ( H ) onsists of subgroups of order p . No w H ats on this set b y onjugation, so the stabilizer of an y E ∈ A p ( H ) is simply N H ( E ) . Also, for ea h E ∈ A p ( H ) , w e see that the set of E - xed p oin ts A p ( H ) E is simply the set { E } . W e then ha v e the follo wing short exat sequene of F p H -mo dules: J F p A p ( H ) ε ։ F p , where ε denotes the augmen tation map, and w e see that for ea h E ∈ A p ( H ) , J is free as an F p E -mo dule with basis { E ′ − E : E ′ ∈ A p ( H ) } . Therefore, if K is an y nite subgroup of H , w e see that J restrited to K is an F p K -mo dule su h that its restrition to ev ery elemen tary ab elian p -subgroup of K is free. It then follo ws from Chouinard's Theorem [6 ℄ that J is pro jetiv e as an F p K -mo dule. As this holds for ev ery nite subgroup K of H , it follo ws from Lemma 7.2 that J has nite pro jetiv e dimension o v er F p H . An argumen t similar to the pro of of Theorem 3.2 then sho ws that N H ( P ) has ohomology almost ev erywhere nitary o v er F p . Hene, C H ( P ) ∼ = P × C N ( P ) 22 MAR TIN HAMIL TON has ohomology almost ev erywhere nitary o v er F p , and b y Lemma 6.2 , C N ( P ) is of t yp e FP ∞ o v er F p . Finally , as cd F p C N ( P ) ≤ cd F p N < ∞ , w e see that C N ( P ) is of t yp e FP o v er F p , and the result no w follo ws. W e an no w answ er Leary and Nuinkis' question. This is a v ariation on Theorem 6.4 : Theorem E. L et G b e a gr oup of typ e VFP over F p , and P b e a p - sub gr oup of G . Then the entr alizer C G ( P ) of P is also of typ e VFP over F p . Pr o of. If P is trivial, then the result is immediate. Assume, therefore, that P has order p k , where k ≥ 1 . W e pro eed b y indution on k : If k = 1 , then the result follo ws from Prop osition 7.3. Supp ose no w that k ≥ 2 . Cho ose a subgroup E ≤ ζ ( P ) of order p . Then C G ( E ) is of t yp e VFP o v er F p b y Prop osition 7.3, and so C G ( E ) has a normal subgroup N of nite index whi h is of t yp e FP o v er F p . Let H := N E . Then C G ( E ) / E has the subgroup H /E of nite index, with H /E ∼ = N of t yp e FP o v er F p , and so C G ( E ) / E is of t yp e VFP o v er F p . By indution, the en tralizer of P /E in C G ( E ) / E is of t yp e VFP o v er F p . Hene the normalizer of P /E in C G ( E ) / E , whi h is ( N G ( P ) ∩ C G ( E ) ) /E , is of t yp e FP ∞ o v er F p , and b y Prop osition 2 . 7 in [2℄ w e see that N G ( P ) ∩ C G ( E ) is of t yp e FP ∞ o v er F p . Then, as C G ( P ) ≤ N G ( P ) ∩ C G ( E ) ≤ N G ( P ) , w e onlude that C G ( P ) is also of t yp e FP ∞ o v er F p . No w, as G is of t yp e VFP o v er F p , it has a subgroup S of nite index whi h is of t yp e FP o v er F p . Therefore, C S ( P ) is of t yp e FP ∞ o v er F p , and as cd F p C S ( P ) ≤ cd F p S < ∞ , w e see that C S ( P ) is of t yp e FP o v er F p , and the result no w follo ws. 8. Gr oups Possessing a Finite Dimensional Model f or E G In this short setion w e onsider groups p ossessing a nite dimen- sional mo del for the lassifying spae E G for prop er ations. The pro of of Theorem 3.8 generalizes immediately to giv e us the follo wing: Prop osition F. L et G b e a gr oup whih p ossesses a nite dimensional mo del for the lassifying sp a e E G for pr op er ations. If (i) G has nitely many onjugay lasses of nite sub gr oups; and WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 23 (ii) The normalizer of every non-trivial nite sub gr oup of G has ohomolo gy almost everywher e nitary, Then G has ohomolo gy almost everywher e nitary. Ho w ev er, the on v erse is false. In fat, it is false ev en for the sub lass of groups of nite virtual ohomologial dimension, as w e shall no w sho w. W e need the follo wing result of Leary (Theorem 20 in [12℄): Prop osition 8.1. L et Q b e a nite gr oup not of prime p ower or der. Then ther e is a gr oup H of typ e F and a gr oup G = H ⋊ Q suh that G ontains innitely many onjugay lasses of sub gr oups isomorphi to Q and nitely many onjugay lasses of other nite sub gr oups. As H is of t yp e F , it has a nite Eilen b ergMa Lane spae, sa y Y . As the univ ersal o v er e Y of Y is on tratible, its augmen ted ellular hain omplex is an exat sequene of Z H -mo dules: 0 → C n ( e Y ) → · · · → C 0 ( e Y ) → Z → 0 , and as Y has only nitely man y ells in ea h dimension, w e see that ea h C k ( e Y ) is nitely generated. Hene, w e see that H has nite ohomologial dimension, and is of t yp e FP ∞ . Therefore, G is a group of nite virtual ohomologial dimension whi h is of t yp e FP ∞ , and hene has ohomology almost ev- erywhere nitary , but G do es not ha v e nitely man y onjugay lasses of nite subgroups, whi h giv es us a oun ter-example to the on v erse of Prop osition F ab o v e. 9. Pr oof of Lemma 1.2 9.1. Pro of of (i) ⇒ (ii). Prop osition 9.1. L et G b e a lo al ly (p olyyli-by-nite) gr oup suh that ther e is a nite dimensional mo del for E G and ther e is a b ound on the or ders of the nite sub gr oups of G . Then G has nite virtual ohomolo gi al dimension. Pr o of. Let X b e a nite dimensional mo del for E G , and let r = dim X . Then, for ea h k , C k ( X ) is a p erm utation mo dule, C k ( X ) ∼ = M σ ∈ Σ k Z [ G σ \ G ] , where Σ k is a set of G -orbit represen tativ es of k -ells in X , and G σ is the stabilizer of σ , so in partiular ea h G σ is nite. Then C k ( X ) ⊗ Q ∼ = M σ ∈ Σ k Q ⊗ Q G σ Q G, 24 MAR TIN HAMIL TON and as Q G σ is semisimple, Q is a pro jetiv e Q G σ -mo dule. Hene, Q ⊗ Q G σ Q G is a pro jetiv e Q G -mo dule, and so C k ( X ) ⊗ Q is a pro jetiv e Q G -mo dule. W e then ha v e that 0 → C r ( X ) ⊗ Q → · · · → C 0 ( X ) ⊗ Q → Q → 0 is a pro jetiv e resolution of the trivial Q G -mo dule, and hene that G has nite rational ohomologial dimension. No w, aording to Hillman and Linnell [9℄, the Hirs h length of an elemen tary amenable group is b ounded ab o v e b y its rational ohomo- logial dimension, so w e onlude that G has nite Hirs h length. Next, let τ ( G ) denote the unique largest lo ally nite normal sub- group of G . As there is a b ound on the orders of the nite subgroups of G , w e see that τ ( G ) m ust b e nite. Then, as G is an elemen tary amenable group of nite Hirs h length, it follo ws from a result of W ehrfritz [18 ℄ that G/τ ( G ) has a p oly-(torsion- free ab elian) harateristi subgroup of nite index. Hene, G has a p oly-(torsion-free ab elian) harateristi subgroup, sa y S , of nite index. W e see that S has nite Hirs h length, and hene nite o- homologial dimension. W e then onlude that G has nite virtual ohomologial dimension, as required. 9.2. Pro of of (ii) ⇒ (iii). W e b egin b y pro ving the follo wing lemma: Lemma 9.2. L et Q b e a nite gr oup, and A b e a Z -torsion-fr e e Z Q - mo dule of nite Hirsh length. Then H 1 ( Q, A ) is nite. Pr o of. As Q is nite, H 1 ( Q, A ) has exp onen t dividing the order of Q (Corollary 10 . 2 I I I in [5℄). W e ha v e the follo wing short exat sequene: A | Q | A π ։ A/ | Q | A. P assing to the long exat sequene in ohomology , w e obtain the fol- lo wing monomorphism: H 1 ( Q, A ) π ∗ H 1 ( Q, A/ | Q | A ) . No w, as A/ | Q | A has nite exp onen t and nite Hirs h length, it is nite. It then follo ws that H 1 ( Q, A ) is nite. Next, note that all groups of nite virtual ohomologial dimension p ossess a nite dimensional mo del for E G (Exerise VI I I. 3 in [5 ℄), so it sues to pro v e the follo wing: Prop osition 9.3. L et G b e a lo al ly (p olyyli-by-nite) gr oup of - nite virtual ohomolo gi al dimension. Then G has nitely many on- jugay lasses of nite sub gr oups. WHEN IS GR OUP COHOMOLOGY FINIT AR Y? 25 Pr o of. As G has nite virtual ohomologial dimension, it m ust ha v e a b ound on the orders of its nite subgroups. Therefore, the same argumen t as in the previous subsetion sho ws that G is a p oly-(torsion- free ab elian)-b y-nite group of nite Hirs h length. W e pro eed b y indution on the Hirs h length h ( G ) of G . If h ( G ) =1, then G has a torsion-free ab elian normal subgroup A of nite Hirs h length su h that G/ A = Q is nite. There is a 1-1 orresp ondene b et w een the onjugay lasses of omplemen ts to A in G and H 1 ( Q, A ) (Result 11.1.3 in [ 16 ℄). Therefore, b y Lemma 9.2, w e see that G has nitely man y onjugay lasses of nite subgroups. Supp ose h ( G ) > 1 . W e kno w that G has a torsion-free ab elian normal subgroup A of nite Hirs h length. As h ( G ) > h ( G/ A ) , w e see b y indution that G/ A has nitely man y onjugay lasses of nite subgroups. Let F b e a nite subgroup of G , so AF lies in one of nitely man y onjugay lasses, sa y those represen ted b y AK 1 , . . . , AK m . Then, as ea h H 1 ( K i , A ) is nite, there are only nitely man y onjugay lasses of omplemen ts to A in AK i , and F m ust lie in one of those. 9.3. Pro of of (iii) ⇒ (i). Let G b e a group with nitely man y onjugay lasses of nite sub- groups. Then it is lear that there m ust b e a b ound on the orders of its nite subgroups. Referenes 1. Ji°í A dámek and Ji°í Rosi ký, L o al ly pr esentable and a essible ate gories , Lon- don Mathematial So iet y Leture Note Series, v ol. 189, Cam bridge Univ ersit y Press, Cam bridge, 1994. 2. Rob ert Bieri, Homolo gi al dimension of disr ete gr oups , seond ed., Queen Mary College Mathematial Notes, Queen Mary College Departmen t of Pure Mathematis, London, 1981. 3. Glen E. Bredon, Equivariant ohomolo gy the ories , Leture Notes in Mathemat- is, No. 34, Springer-V erlag, Berlin, 1967. 4. Kenneth S. Bro wn, Homolo gi al riteria for niteness , Commen t. Math. Helv. 50 (1975), 129135. 5. , Cohomolo gy of gr oups , Graduate T exts in Mathematis, v ol. 87, Springer-V erlag, New Y ork, 1982. 6. Leo G. Chouinard, Pr oje tivity and r elative pr oje tivity over gr oup rings , J. Pure Appl. Algebra 7 (1976), no. 3, 287302. 7. Jonathan Corni k and P eter H. 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Dep ar tment of Ma thema tis, University of Glasgo w, University Gar- dens, Glasgo w G12 8QW, United Kingdom E-mail addr ess : m.hamiltonmath s.g la .a .u k
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