Subgroup posets, Bredon cohomology and equivariant Euler characteristics

Martínez-Pérez, Conchita

doi:10.1090/S0002-9947-2013-05781-9

Subgroup posets, Bredon cohomology and equivariant Euler characteristics
HTML articles powered by AMS MathViewer

by Conchita Martínez-Pérez PDF

Trans. Amer. Math. Soc. 365 (2013), 4351-4370 Request permission

Abstract:

For discrete groups $\Gamma$ with a bound on the order of their finite subgroups, we construct Bredon projective resolutions of the trivial module in terms of projective covers of the chain complex associated to the poset of finite subgroups. We use this to give new results on dimensions of $\underline {\operatorname {E}}\Gamma$ and to reprove that for virtually solvable groups, $\underline {\mathrm {cd}}\Gamma =\mathrm {vcd}\Gamma$. We also deduce a formula to compute the equivariant Euler class of $\underline {\operatorname {E}}\Gamma$ for $\Gamma$ virtually solvable of type $\mathrm {FP}_\infty$ and use it to compute orbifold Euler characteristics.

References

A. Adem, M. Klaus. Lectures on orbifolds and group cohomology, Advanced Lectures in Mathematics 16, Transformation Groups and Moduli Spaces of Curves. Higher Education Press, edited by L.Ji and S-T Yau, 2010.
Alejandro Adem, Johann Leida, and Yongbin Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007. MR 2359514, DOI 10.1017/CBO9780511543081
Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671–677. MR 1076708, DOI 10.1016/0393-0440(89)90032-6
D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Kenneth S. Brown, Complete Euler characteristics and fixed-point theory, J. Pure Appl. Algebra 24 (1982), no. 2, 103–121. MR 651839, DOI 10.1016/0022-4049(82)90008-1
Frank Connolly and Tadeusz Koźniewski, Finiteness properties of classifying spaces of proper $\Gamma$-actions, J. Pure Appl. Algebra 41 (1986), no. 1, 17–36. MR 844461, DOI 10.1016/0022-4049(86)90097-6
Jonathan Cornick and Peter H. Kropholler, Homological finiteness conditions for modules over group algebras, J. London Math. Soc. (2) 58 (1998), no. 1, 49–62. MR 1666074, DOI 10.1112/S0024610798005729
Beno Eckmann, Amenable groups and Euler characteristic, Comment. Math. Helv. 67 (1992), no. 3, 383–393. MR 1171301, DOI 10.1007/BF02566509
P. H. Kropholler, C. Martinez-Pérez, and B. E. A. Nucinkis, Cohomological finiteness conditions for elementary amenable groups, J. Reine Angew. Math. 637 (2009), 49–62. MR 2599081, DOI 10.1515/CRELLE.2009.090
Peter H. Kropholler and Guido Mislin, Groups acting on finite-dimensional spaces with finite stabilizers, Comment. Math. Helv. 73 (1998), no. 1, 122–136. MR 1610595, DOI 10.1007/s000140050048
P. H. Kropholler and Olympia Talelli, On a property of fundamental groups of graphs of finite groups, J. Pure Appl. Algebra 74 (1991), no. 1, 57–59. MR 1129129, DOI 10.1016/0022-4049(91)90048-7
M. Langer, W. Lück. On the group cohomology of the semi-direct product $\mathbb {Z}^n\rtimes _\rho \mathbb {Z}_m$ and a conjecture of Adem-Ge-Petrosyan. http://arxiv.org/pdf/1105.4772v1
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
John C. Lennox and Derek J. S. Robinson, The theory of infinite soluble groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2004. MR 2093872, DOI 10.1093/acprof:oso/9780198507284.001.0001
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. $L^2$-Invariants: Theory and Applications to Geometry and K-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer, 2002.
Conchita Martínez-Pérez, A bound for the Bredon cohomological dimension, J. Group Theory 10 (2007), no. 6, 731–747. MR 2364823, DOI 10.1515/JGT.2007.053
Conchita Martinez-Pérez and Brita E. A. Nucinkis, Virtually soluble groups of type $\rm FP_\infty$, Comment. Math. Helv. 85 (2010), no. 1, 135–150. MR 2563683, DOI 10.4171/CMH/190
Peter Symonds, The Bredon cohomology of subgroup complexes, J. Pure Appl. Algebra 199 (2005), no. 1-3, 261–298. MR 2134305, DOI 10.1016/j.jpaa.2004.12.010
Jacques Thévenaz, Equivariant $K$-theory and Alperin’s conjecture, J. Pure Appl. Algebra 85 (1993), no. 2, 185–202. MR 1207508, DOI 10.1016/0022-4049(93)90052-U
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136