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Centralizers in Houghton’s Groups

Published online by Cambridge University Press:  05 January 2015

Simon St. John-Green
Simon St. John-Green*
Affiliation:
School of Mathematics, University of Southampton, University Road, Southampton SO17 1BJ, UK ([email protected])
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Abstract

We calculate the centralizers of elements, finite subgroups and virtually cyclic subgroups of Houghton’s group Hn. We discuss various Bredon (co)homological finiteness conditions satisfied by Hn including the Bredon (co)homological dimension and FPn conditions, which are analogues of the ordinary cohomological dimension and FPn conditions, respectively.

Keywords

Houghton’s groupcentralizersfiniteness conditionsPrimary 20J0520E07
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 769 - 785
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1.Bieri, R., Homological dimension of discrete groups, Queen Mary College Mathematics Notes (Queen Mary College, London, 1976).Google Scholar
2.Bleak, C., Bowman, H., Gordon, A., Graham, G., Hughes, J., Matucci, F. and Sapir, J., Centralizers in R. Thompson’s group V n, preprint (arXiv.org/abs/1107.0672, 2011).Google Scholar
3.Brady, N., Leary, I. J. and Nucinkis, B. E. A., On algebraic and geometric dimensions for groups with torsion, J. Lond. Math. Soc. 64(2 (2001), 489–500.Google Scholar
4.Brown, K. S., Cohomology of groups (Springer, 1982).Google Scholar
5.Brown, K. S., Finiteness properties of groups, J. Pure Appl. Alg. 44(1–3) (1987), 45–75.CrossRefGoogle Scholar
6.Brown, K. S. and Geoghegan, R., An infinite-dimensional torsion-free FP∞ group, Invent. Math. 77(2 (1984), 367–381.CrossRefGoogle Scholar
7.Degrijse, D. and Petrosyan, N., Geometric dimension of groups for the family of virtually cyclic subgroups, J. Topolog. 7(3 (2014), 697–726.CrossRefGoogle Scholar
8.Dunwoody, M. J., Accessibility and groups of cohomological dimension one, Proc. Lond. Math. Soc. s3–38(2 (1979), 193–215.Google Scholar
9.Gandini, G., Bounding the homological finiteness length, Bull. Lond. Math. Soc. 44(6 (2012), 1209–1214.Google Scholar
10.Hillman, J. A. and Linnell, P. A., Elementary amenable groups of finite Hirsch length are locally-finite by virtually-solvable, J. Austral. Math. Soc. A 52(2 (1992), 237–241.CrossRefGoogle Scholar
11.Houghton, C. H., The first cohomology of a group with permutation module coefficients, Arch. Math. 31(1 (1978), 254–258.CrossRefGoogle Scholar
12.Juan-Pineda, D. and Leary, I. J., On classifying spaces for the family of virtually cyclic subgroups, in Recent developments in algebraic topology, Contemporary Mathematics, Volume 407, pp. 135–145 (American Mathematical Society, Providence, RI, 2006).Google Scholar
13.Kochloukova, D. H., Martínez-Pèrez, C. and B. Nucinkis, E. A., Centralisers of finite subgroups in soluble groups of type FPn, Forum Math. 23(1 (2011), 99–115.Google Scholar
14.Kropholler, P. H., Martínez-Peèrez, C. and Nucinkis, B. E. A., Cohomological finiteness conditions for elementary amenable groups, J. Reine Angew. Math. 2009(637 (2009), 49–62.CrossRefGoogle Scholar
15.Leary, I. J. and Nucinkis, B. E. A., Bounding the orders of finite subgroups, Publ. Mat. 45(1 (2001), 259–264.Google Scholar
16.Lück, W., Transformation groups and algebraic K-theory (Springer, 1989).Google Scholar
17.Lück, W., The type of the classifying space for a family of subgroups, J. Pure Appl. Alg. 149(2 (2000), 177–203.CrossRefGoogle Scholar
18.Lück, W., Survey on classifying spaces for families of subgroups, in Infinite groups: geometric, combinatorial and dynamical aspects, Progress in Mathematics, Volume 248, pp. 269–322 (Birkhäser, 2003).Google Scholar
19.Lück, W. and Meintrup, D., On the universal space for group actions with compact isotropy, in Geometry and Topology: Aarhus, Conference on Geometry and Topology, August 10–16, Aarhus University, Aarhus, Denmark, 1998, Contemporary Mathematics, Volume 258, pp. 293–305 (American Mathematical Society, Providence, RI, 2000).Google Scholar
20.Lück, W. and Weiermann, M., On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8(2 (2012), 497–555.Google Scholar
21.Martinez-Pérez, C. and Nucinkis, B. E. A., Virtually soluble groups of type FP∞, Comment. Math. Helv. 85(1 (2010), 135–150.CrossRefGoogle Scholar
22.Martinéz-Pérez, C. and Nucinkis, B. E. A., Bredon cohomological finiteness conditions for generalisations of Thompson groups, Groups Geom. Dynam. 7(4 (2013), 931–959.CrossRefGoogle Scholar
23.Robinson, D., A course in the theory of groups (Springer, 1996).Google Scholar
24.Rotman, J. J., An introduction to the theory of groups (Springer, 1995).CrossRefGoogle Scholar