Groups, Geometry, and Dynamics

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Volume 7, Issue 4, 2013, pp. 931–959
DOI: 10.4171/GGD/211

Published online: 2013-11-20

Bredon cohomological finiteness conditions for generalisations of Thompson groups

Conchita Martínez-Pérez[1] and Brita E. A. Nucinkis[2]

(1) Universidad de Zaragoza, Spain
(2) Royal Holloway University of London, Egham, UK

We define a family of groups that generalises Thompson’s groups $T$ and $G$, and also those of Higman, Stein and Brin. For groups in this family we describe centralisers of finite subgroups and show that for a given finite subgroup $Q$ there are finitely many conjugacy classes of finite subgroups isomorphic to $Q$. We consider groups of type quasi-$\underline{\rm F}_\infty$. This is a property slightly weaker than possessing a finite type model for the classifying space for proper actions $\underline{E}G$. We give criteria for the $T$ versions of our groups to be of type quasi-$\underline{\rm F}_\infty$. We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.

Keywords: Bredon cohomology, Thompson groups, finiteness properties

Martínez-Pérez Conchita, Nucinkis Brita E. A.: Bredon cohomological finiteness conditions for generalisations of Thompson groups. Groups Geom. Dyn. 7 (2013), 931-959. doi: 10.4171/GGD/211