On Bredon homology of elementary amenable groups

Flores, Ramón; Nucinkis, Brita

doi:10.1090/S0002-9939-06-08565-0

On Bredon homology of elementary amenable groups
HTML articles powered by AMS MathViewer

by Ramón J. Flores and Brita E. A. Nucinkis PDF

Proc. Amer. Math. Soc. 135 (2007), 5-11 Request permission

Abstract:

We show that for elementary amenable groups the Hirsch length is equal to the Bredon homological dimension. This also implies that countable elementary amenable groups admit a finite-dimensional model for $\underline {E}G$ of dimension less than or equal to the Hirsch length plus one. Some remarks on groups of type ${\operatorname {FP}}_{\infty }$ are also made.

References

Gilbert Baumslag and Robert Bieri, Constructable solvable groups, Math. Z. 151 (1976), no. 3, 249–257. MR 422422, DOI 10.1007/BF01214937
Robert Bieri, Homological dimension of discrete groups, Queen Mary College Mathematics Notes, Queen Mary College, Mathematics Department, London, 1976. MR 0466344
Noel Brady, Ian J. Leary, and Brita E. A. Nucinkis, On algebraic and geometric dimensions for groups with torsion, J. London Math. Soc. (2) 64 (2001), no. 2, 489–500. MR 1853466, DOI 10.1112/S002461070100240X
Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Warren Dicks, Peter H. Kropholler, Ian J. Leary, and Simon Thomas, Classifying spaces for proper actions of locally finite groups, J. Group Theory 5 (2002), no. 4, 453–480. MR 1931370, DOI 10.1515/jgth.2002.016
Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050, DOI 10.1515/9783110858372.312
M. J. Dunwoody, Accessibility and groups of cohomological dimension one, Proc. London Math. Soc. (3) 38 (1979), no. 2, 193–215. MR 531159, DOI 10.1112/plms/s3-38.2.193
D. Gildenhuys and R. Strebel, On the cohomological dimension of soluble groups, Canad. Math. Bull. 24 (1981), no. 4, 385–392. MR 644526, DOI 10.4153/CMB-1981-060-x
D. Gildenhuys and R. Strebel, On the cohomology of soluble groups. II, J. Pure Appl. Algebra 26 (1982), no. 3, 293–323. MR 678526, DOI 10.1016/0022-4049(82)90112-8
Jonathan A. Hillman, Elementary amenable groups and $4$-manifolds with Euler characteristic $0$, J. Austral. Math. Soc. Ser. A 50 (1991), no. 1, 160–170. MR 1094067
J. A. Hillman and P. A. Linnell, Elementary amenable groups of finite Hirsch length are locally-finite by virtually-solvable, J. Austral. Math. Soc. Ser. A 52 (1992), no. 2, 237–241. MR 1143191
P. H. Kropholler, Cohomological dimension of soluble groups, J. Pure Appl. Algebra 43 (1986), no. 3, 281–287. MR 868988, DOI 10.1016/0022-4049(86)90069-1
P. H. Kropholler, Soluble groups of type $(\textrm {FP})_\infty$ have finite torsion-free rank, Bull. London Math. Soc. 25 (1993), no. 6, 558–566. MR 1245082, DOI 10.1112/blms/25.6.558
P. H. Kropholler, P. A. Linnell, and J. A. Moody, Applications of a new $K$-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988), no. 3, 675–684. MR 964842, DOI 10.1090/S0002-9939-1988-0964842-0
Ian J. Leary and Brita E. A. Nucinkis, Some groups of type $VF$, Invent. Math. 151 (2003), no. 1, 135–165. MR 1943744, DOI 10.1007/s00222-002-0254-7
Wolfgang Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989. Mathematica Gottingensis. MR 1027600, DOI 10.1007/BFb0083681
Wolfgang Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000), no. 2, 177–203. MR 1757730, DOI 10.1016/S0022-4049(98)90173-6
W. Lück, Survey on Classifying Spaces for Families of Subgroups, Preprintreihe SFB 478 - Geometrische Strukturen in der Mathematik 308 (2004).
Conchita Martínez-Pérez, A spectral sequence in Bredon (co)homology, J. Pure Appl. Algebra 176 (2002), no. 2-3, 161–173. MR 1933713, DOI 10.1016/S0022-4049(02)00154-8
Guido Mislin, On the classifying space for proper actions, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 263–269. MR 1851258
Guido Mislin, Equivariant $K$-homology of the classifying space for proper actions, Proper group actions and the Baum-Connes conjecture, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003, pp. 1–78. MR 2027169
Brita E. A. Nucinkis, Is there an easy algebraic characterisation of universal proper $G$-spaces?, Manuscripta Math. 102 (2000), no. 3, 335–345. MR 1777524, DOI 10.1007/s002291020335
Brita E. A. Nucinkis, On dimensions in Bredon homology, Homology Homotopy Appl. 6 (2004), no. 1, 33–47. MR 2061566
Urs Stammbach, On the weak homological dimension of the group algebra of solvable groups, J. London Math. Soc. (2) 2 (1970), 567–570. MR 263927, DOI 10.1112/jlms/2.Part_{3}.567
B. A. F. Wehrfritz, On elementary amenable groups of finite Hirsch number, J. Austral. Math. Soc. Ser. A 58 (1995), no. 2, 219–221. MR 1323993