Finitary Group Cohomology and Group Actions on Spheres

FINIT AR Y GR OUP COHOMOLOGY AND GR OUP A CTIONS ON SPHERES MAR TIN HAMIL TON Abstra t. W e sho w that if G is an innitely generated lo ally (p olyyli-b y-nite) group with ohomology almost ev erywhere nitary , then ev ery nite subgroup of G ats freely and orthogo- nally on some sphere. 1. Intr odution In [3℄ the question of whi h lo ally (p olyyli-b y-nite) groups ha v e ohomology almost ev erywhere nitary w as onsidered. Reall that a funtor is nitary if it preserv es ltered olimits (see 6.5 in [7℄; also 3.18 in [1℄). The n th ohomology of a group G is a funtor H n ( G, − ) from the ategory of Z G -mo dules to the ategory of ab elian groups. If G is a lo ally (p olyyli-b y-nite) group, then Theorem 2.1 in [ 5℄ sho ws that the nitary set F ( G ) := { n ∈ N : H n ( G, − ) is nitary } is either onite or nite. If F ( G ) is onite, w e sa y that G has  ohomolo gy almost everywher e nitary , and if F ( G ) is nite, w e sa y that G has  ohomolo gy almost everywher e innitary . W e pro v ed the follo wing results ab out lo ally (p olyyli-b y-nite) groups with ohomology almost ev erywhere nitary in [ 3℄: Theorem 1.1. L et G b e a lo  al ly (p olyyli-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 is nitely gener ate d. Corollary 1.2. L et G b e a lo  al ly (p olyyli-by-nite) gr oup with  o- homolo gy almost everywher e nitary. Then every sub gr oup of G also has  ohomolo gy almost everywher e nitary. 2000 Mathematis Subje t Classi ation. 20J06 18A22 20E34. Key wor ds and phr ases. ohomology of groups, nitary funtors, group ations on spheres. 1 2 MAR TIN HAMIL TON Reall (see, for example, [10 ℄) that a nite group ats freely and orthogonally on some sphere if and only if ev ery subgroup of order pq , where p and q are prime, is yli. In this pap er, w e pro v e the follo wing result: Theorem A. L et G b e an innitely gener ate d lo  al ly (p olyyli-by- nite) gr oup with  ohomolo gy almost everywher e nitary. Then every nite sub gr oup of G ats fr e ely and ortho gonal ly on some spher e. Note that w e annot remo v e the "innitely generated" restrition; as, for example, ev ery nite group is of t yp e FP ∞ and so has n th ohomology funtors nitary for all n , b y a result of Bro wn (Corollary to Theorem 1 in [2℄). 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 the help he has giv en me with this pap er. I w ould also lik e to thank F rank Quinn for suggesting that a result lik e Theorem A should b e true. 2. Pr oof The follo wing Prop osition sets the sene for pro ving Theorem A: Prop osition 2.1. L et G b e a lo  al ly (p olyyli-by-nite) gr oup with  ohomolo gy almost everywher e nitary. Then G has a har ateristi sub gr oup S of nite index, suh that S is torsion-fr e e soluble of nite Hirsh length. Pr o of. By Theorem 2.1 in [ 5℄ w e kno w that there is a nite-dimensional on tratible G -CW-omplex X on whi h G ats with nite isotrop y groups, and that there is a b ound on the orders of the nite subgroups of G . W e kno w that the rational ohomologial dimension of G is b ounded ab o v e b y the dimension of X (see, for example, [6℄), so G has nite rational ohomologial dimension. Reall that the lass of elemen tary amenable groups is the lass generated from the nite groups and Z b y the op erations of extension and inreasing union (see, for example, [4 ℄), so G is elemen tary amenable. A ording to [4℄, the Hirs h length of an elemen tary amenable group is b ounded ab o v e b y its rational ohomologial dimension, so G has nite Hirs h length. Let τ ( G ) denote the join of the lo ally nite normal subgroups of G . As there is a b ound on the orders of the nite subgroups of G , this implies that τ ( G ) is nite. Replaing G with G/τ ( G ) , w e ma y assume that τ ( G ) = 1 . FINIT AR Y GR OUP COHOMOLOGY AND GR OUP A CTIONS ON SPHERES 3 No w G is an elemen tary amenable group of nite Hirs h length, so it follo ws from a minor extension of a theorem b y Mal'ev (see W ehrfritz's pap er [12 ℄) that G/τ ( G ) = G has a p oly (torsion-free ab elian)  hara- teristi subgroup of nite index.  Before pro ving Theorem A, w e need four Lemmas: Lemma 2.2. L et Q b e a non-yli gr oup of or der pq , wher e p and q ar e prime, and let A b e a Z -torsion-fr e e Z Q -mo dule suh that the gr oup A ⋊ Q has  ohomolo gy almost everywher e nitary. Then A is nitely gener ate d. Pr o of. W e write G := A ⋊ Q . F or an y K ≤ Q w e write b K for the elemen t of Z Q giv en b y b K := X k ∈ K k . Notie that b K .A is on tained in the set of K -in v arian t elemen ts A K of A . There are t w o ases to onsider: If Q is ab elian, then p = q and Q has p + 1 subgroups E 0 , . . . , E p of order p . W e ha v e the follo wing equation in Z Q : p X i =0 c E i = b Q + p. 1 , so it follo ws that for an y a ∈ A p.a = p X i =0 c E i .a − b Q.a ∈ p X i =0 A E i + A Q and hene p.A ⊆ p X i =0 A E i + A Q . If K is non-trivial, then it follo ws from Theorem 1.1 that N G ( K ) is nitely generated. Then, as A K ≤ N G ( K ) , it follo ws that A K is also nitely generated. Hene w e see that p.A is nitely generated, and as A is torsion-free, w e onlude that A is nitely generated. If Q is non-ab elian, then p 6 = q , and without loss of generalit y w e ma y assume that p < q . Then Q has one subgroup F of order q and q subgroups H 0 , . . . , H q − 1 of order p . W e ha v e the follo wing equation in 4 MAR TIN HAMIL TON Z Q : q − 1 X i =0 c H i + b F = b Q + q . 1 and the pro of on tin ues as ab o v e.  Reall (see, for example, 10.4 in [8℄) that a group G is upp er-nite if and only if ev ery nitely generated homomorphi image of G is - nite. The lass of upp er-nite groups is losed under extensions and homomorphi images. Also reall (see 10.4 in [8℄) that the upp er-nite r adi al of a group G is the subgroup generated b y all of its upp er-nite normal subgroups, and is itself upp er-nite. Lemma 2.3. L et A and B b e ab elian gr oups. If A is upp er-nite, then A ⊗ B is upp er-nite. Pr o of. If b ∈ B , then A ⊗ b is a homomorphi image of A and hene is upp er-nite. Then as A ⊗ B is generated b y all the A ⊗ b it is also upp er-nite.  Lemma 2.4. L et G b e an upp er-nite nilp otent gr oup. Then its derive d sub gr oup G ′ is also upp er-nite. Pr o of. As G is upp er-nite, it follo ws that G/G ′ is also upp er-nite. As G is a nilp oten t group, it has a nite lo w er en tral series G = γ 1 ( G ) ≥ γ 2 ( G ) ≥ · · · ≥ γ k ( G ) = 1 , where γ 2 ( G ) = G ′ . F or ea h i there is an epimorphism G/G ′ ⊗ · · · ⊗ G/G ′ | {z } i ։ γ i ( G ) /γ i +1 ( G ) , and as G/G ′ ⊗ · · · ⊗ G/G ′ | {z } i is upp er-nite, from Lemma 2.3 , w e see that ea h γ i ( G ) /γ i +1 ( G ) is upp er-nite. Then, as the lass of upp er-nite groups is losed under extensions, w e onlude that G ′ is also upp er- nite.  Lemma 2.5. L et G b e a torsion-fr e e nilp otent gr oup of nite Hirsh length. If the  entr e ζ ( G ) of G is nitely gener ate d, then G is nitely gener ate d. Pr o of. Let K b e the upp er-nite radial of G . As G is torsion-free nilp oten t of nite Hirs h length, it is a sp eial ase of Lemma 10.45 in [8 ℄ that G/K is nitely generated. Supp ose that K 6 = 1 . FINIT AR Y GR OUP COHOMOLOGY AND GR OUP A CTIONS ON SPHERES 5 F ollo wing an argumen t of Robinson (Lemma 10.44 in [8℄) w e see that for ea h g ∈ G , [ K, g ] K ′ /K ′ is a homomorphi image of K , and so is upp er-nite, so therefore [ K, G ] / K ′ is upp er-nite. Then, as K ′ is upp er-nite, from Lemma 2.4 , w e see that [ K, G ] is also upp er-nite. Similarly , w e see b y indution that [ K, m G ] = [ K , G, . . . , G | {z } m ] is upp er- nite. Cho ose the largest m su h that [ K, m G ] 6 = 1 . Then [ K, m G ] ⊆ ζ ( G ) , so [ K, m G ] is nitely generated, and hene nite. Then, as G is torsion- free, w e see that [ K, m G ] = 1 , whi h is a on tradition. Therefore, K = 1 , and so G is nitely generated.  W e an no w pro v e Theorem A. Pr o of of The or em A. Let G b e an innitely generated lo ally (p olyyli- b y-nite) group with ohomology almost ev erywhere nitary . It follo ws from Prop osition 2.1 that G has a  harateristi subgroup S of nite index su h that S is torsion-free soluble of nite Hirs h length. Supp ose that not ev ery subgroup of G ats freely and orthogonally on some sphere, so there is a non-yli subgroup Q of order pq , where p and q are prime. As S is a torsion-free soluble group of nite Hirs h length, it is linear o v er the rationals (see, for example, [11℄), so b y a result of Gruen b erg (see Theorem 8.2 of [11 ℄) the Fitting subgroup F := Fitt( S ) of S is nilp oten t. No w the en tre ζ ( F ) of F is a  harateristi subgroup of G , so w e an onsider the group ζ ( F ) Q . It then follo ws from Corollary 1.2 that ζ ( F ) Q has ohomology almost ev erywhere nitary . Then, b y Lemma 2.2 , w e see that ζ ( F ) is nitely generated. It then follo ws from Lemma 2.5 that F is nitely generated. No w, let K b e the subgroup of S on taining F su h that K/ F = τ ( S/F ) . As S is linear o v er Q , w e see that S/F is also linear o v er Q , and as lo ally nite Q -linear groups are nite (see, for example, Theorem 9.33 in [11 ℄), w e onlude that K/ F is nite. An argumen t of Zassenhaus in 15.1.2 of [9℄ sho ws that S/K is maximal ab elian-b y- nite; that is, rystallographi. Hene S/F is nitely generated, so w e onlude that S is nitely generated, a on tradition.  Referenes 1. Ji°í A dámek and Ji°í Rosi ký, L o  al ly pr esentable and a  essible  ate gories , Lon- don Mathematial So iet y Leture Note Series, v ol. 189, Cam bridge Univ ersit y Press, Cam bridge, 1994. 6 MAR TIN HAMIL TON 2. Kenneth S. Bro wn, Homolo gi al riteria for niteness , Commen t. Math. Helv. 50 (1975), 129135. 3. Martin Hamilton, When is gr oup  ohomolo gy nitary? , (Preprin t, Univ ersit y of Glasgo w 2007). 4. J. A. Hillman and P . A. Linnell, Elementary amenable gr oups of nite Hirsh length ar e lo  al ly-nite by virtual ly-solvable , J. Austral. Math. So . Ser. A 52 (1992), no. 2, 237241. 5. P eter H. Kropholler, Gr oups with many nitary  ohomolo gy funtors , (Preprin t, Univ ersit y of Glasgo w 2007). 6. P eter H. Kropholler and Guido Mislin, Gr oups ating on nite dimensional sp a es with nite stabilizers , Commen t. Math. Helv. (1998), no. 73, 122136. 7. T om Leinster, Higher op er ads, higher  ate gories , London Mathematial So iet y Leture Note Series, v ol. 298, Cam bridge Univ ersit y Press, Cam bridge, 2004. 8. Derek J. S. Robinson, Finiteness  onditions and gener alize d soluble gr oups, p art 2 , Ergebnisse der Mathematik und ihrere Grenzegebiete, v ol. 63, Springer- V erlag, 1972. 9. Derek John Sott Robinson, A  ourse in the the ory of gr oups , Graduate T exts in Mathematis, v ol. 80, Springer-V erlag, New Y ork-Berlin, 1982. 10. C. B. Thomas and C. T. C. W all, The top olo gi al spheri al sp a e form pr oblem. 1 , Comp ositio Math. 23 (1971), 101114. 11. B. A. F. W ehrfritz, Innite line ar gr oups. An a  ount of the gr oup-the or eti pr op erties of innite gr oups of matri es , Springer-V erlag, New Y ork, 1973, Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76. 12. , On elementary amenable gr oups of nite Hirsh numb er , J. Austral. Math. So . Ser. A 58 (1995), no. 2, 219221. Dep ar tment of Ma thema tis, University of Glasgo w, University Gar- dens, Glasgo w G12 8QW, United Kingdom E-mail addr ess : m.hamiltonmath s.g la .a .u k