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On Bredon homology of elementary amenable groups


Authors: Ramón J. Flores and Brita E. A. Nucinkis
Journal: Proc. Amer. Math. Soc. 135 (2007), 5-11
MSC (2000): Primary 20J05, 18G20
DOI: https://doi.org/10.1090/S0002-9939-06-08565-0
Published electronically: August 16, 2006
MathSciNet review: 2280168
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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.


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  • 1. B. Baumslag and R. Bieri, Constructable solvable groups, Math. Z. 151 (1976), 249-257. MR 0422422 (54:10411)
  • 2. R. Bieri, Homological dimension of discrete groups, Queen Mary College Mathematics Notes, London (1976). MR 0466344 (57:6224)
  • 3. N.P. Brady, I.J. Leary and B.E.A. Nucinkis, On algebraic and geometric dimensions for groups with torsion, J. London Math. Soc. 64 (2) (2001), 489-500. MR 1853466 (2002h:57007)
  • 4. G.E. Bredon, Equivariant cohomology theories, Springer Lecture Notes in Math. 34 (1967). MR 0214062 (35:4914)
  • 5. K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag (1982). MR 0672956 (83k:20002)
  • 6. W. Dicks, I.J. Leary, P.H. Kropholler and S. Thomas, Classifying spaces for proper actions of locally finite groups, J. Group Theory 5 (4) (2002), 453-480. MR 1931370 (2003g:20064)
  • 7. T. tom Dieck, Transformation Groups, de Gruyter Studies in Mathematics 8 (1987). MR 0889050 (89c:57048)
  • 8. M.J. Dunwoody, Accessibility and groups of cohomological dimension 1, Proc. London Math. Soc. 38 (1997), 193-215. MR 0531159 (80i:20024)
  • 9. D. Gildenhuys and R. Strebel, On the cohomological dimension of soluble groups, Cand. Math. Bull. 24 (4), (1981), 385-392. MR 0644526 (83h:20047)
  • 10. D. Gildenhuys and R. Strebel, On the cohomological dimension of soluble groups II, J. Pure Appl. Algebra 26 (1982), 293-323. MR 0678526 (84c:20059)
  • 11. J.A. Hillman, Elementary amenable groups and $ 4$-manifolds with Euler characteristic 0, J. Austral. Math. Soc Ser. A 50 (1) (1991), 160-170. MR 1094067 (92g:20057)
  • 12. 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 (2) (1992), 237-241. MR 1143191 (93b:20067)
  • 13. P.H. Kropholler, Cohomological dimension of soluble groups, J. Pure Appl. Algebra 43 (1986), 281-287. MR 0868988 (88h:20063)
  • 14. P.H. Kropholler, On groups of type $ FP_\infty$, J. Pure Appl. Algebra 90 (1) (1993), 55-67. MR 1246274 (94j:20051b)
  • 15. P.H. Kropholler, P.A. Linnell and J.A. Moody, Applications of a new $ K$-theoretic theorem to soluble groups rings, Proc. Amer. Math. Soc. 104 (3) (1988), 675-684. MR 0964842 (89j:16016)
  • 16. I.J. Leary and B.E.A. Nucinkis, Some groups of type VF, Invent. Math. 151 (1) (2003), 135-165. MR 1943744 (2003k:20086)
  • 17. W. Lück, Transformation groups and algebraic K-theory, Lecture Notes in Mathematics 1408 Springer, Berlin (1989). MR 1027600 (91g:57036)
  • 18. W. Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000), 177-203. MR 1757730 (2001i:55018)
  • 19. W. Lück, Survey on Classifying Spaces for Families of Subgroups, Preprintreihe SFB 478 - Geometrische Strukturen in der Mathematik 308 (2004).
  • 20. C. Martinez-Pérez, A spectral sequence in Bredon (co)homology, J. Pure Appl. Algebra 176 (2002), 161-173. MR 1933713 (2003h:20095)
  • 21. G. Mislin, On the classifying space for proper actions, In cohomological Methods in Homotopy Theory, Progress in Mathematics 196, Birkhäuser (2001). MR 1851258 (2002f:55032)
  • 22. G. Mislin, Equivariant K-homology of the classifying space for proper actions, In Notes on the Advances Course on proper group actions. CRM Barcelona (2001). MR 2027169 (2005e:19008)
  • 23. B.E.A. Nucinkis, Is there an easy algebraic characterisation of universal proper $ G$-spaces?, Manuscripta Math. 102 (2000), 335-345. MR 1777524 (2001h:20075)
  • 24. B.E.A. Nucinkis, On dimensions in Bredon homology, Homology, Homotopy Appl. 6 (1) (2004), 33-47. MR 2061566 (2005c:20092)
  • 25. U. Stammbach, On the weak homological dimension of the group algebra of solvable groups, J. London Math. Soc. 2 (2) (1970), 567-570. MR 0263927 (41:8526)
  • 26. B.A.F. Wehrfritz, On elementary amenable groups of finite Hirsch number, J. Austral. Math. Soc Ser. A 58 (2), (1995), 219-221. MR 1323993 (96a:20050)

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Additional Information

Ramón J. Flores
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Barcelona, E 08193 Bellaterra, Spain
Address at time of publication: Departamento de Estadística, Universidad Carlos III, Campus de Colmen- arejo, 22 28270 Colmenarejo (Madrid), Spain
Email: ramonj@mat.uab.es, rflores@est-econ.uc3m.es

Brita E. A. Nucinkis
Affiliation: School of Mathematics, University of Southampton, Southampton, SO 17 1BJ, United Kingdom
Email: B.E.A.Nucinkis@soton.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-06-08565-0
Keywords: Elementary amenable group, Bredon homology
Received by editor(s): July 20, 2005
Published electronically: August 16, 2006
Additional Notes: This work was partially supported by MCYT grant BFM2001-2035
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2006 American Mathematical Society

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