Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-28T04:44:11.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Finiteness Properties of Some Groups of Local Similarities

Part of: Spaces with richer structures Special aspects of infinite or finite groups Locally compact groups and their algebras

Published online by Cambridge University Press:  10 April 2015

Daniel S. Farley  and
Bruce Hughes
Daniel S. Farley
Affiliation:
Department of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA, (farleyds@muohio.edu)
Bruce Hughes
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA, (bruce.hughes@vanderbilt.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Hughes has defined a class of groups that we call finite similarity structure (FSS) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson’s group V. Guided by previous work on Thompson’s group, we show that many FSS groups are of type F∞. This generalizes work of Ken Brown from the 1980s.

Keywords

finite similarity structureNekrashevych–Röver groupsThompson’s grouptype F∞ultrametriczipper action

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 22D10: Unitary representations of locally compact groups 54E45: Compact (locally compact) metric spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 379 - 402
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Björner, A., Topological methods, in Handbook of combinatorics, Volume 2, Chapter 34, pp. 1819–1872 (Elsevier, 1995).Google Scholar
2.Brown, K. S., Finiteness properties of groups, J. Pure Appl. Alg. 44 (1987), 45–75.CrossRefGoogle Scholar
3.Farley, D. S., Proper isometric actions of Thompson’s groups on Hilbert space, Int. Math. Res. Not. 2003(45) (2003), 2409–2414.CrossRefGoogle Scholar
4.Farley, D. S., Homological and finiteness properties of picture groups, Trans. Am. Math. Soc. 357(9) (2005), 3567–3584.CrossRefGoogle Scholar
5.Geoghegan, R., Topological methods in group theory, Graduate Texts in Mathematics, Volume 243 (Springer, 2008).CrossRefGoogle Scholar
6.Hughes, B., Local similarities and the Haagerup property, Groups Geom. Dyn. 3 (2009), 299–315.CrossRefGoogle Scholar
7.Nekrashevych, V. V., Cuntz-Pimsner algebras of group actions, J. Operat. Theory 52(2)(2004), 223–249.Google Scholar
8.Röver, C. E., Constructing finitely presented simple groups that contain Grigorchuk groups, J. Alg. 220(1) (1999), 284–313.CrossRefGoogle Scholar