Bredon cohomological finiteness conditions for generalisations of Thompson's groups

Abstract

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 descrine 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 use this to show that whenever defined, the T versions of these groups have a slightly weaker property, quasi-${\underline{F}}_{\infty }$, to that of a group possessing a finite type model for the classifying space for proper actions $\underline{E}G$ if and only if they posses finite type models for the ordinary classifying space. We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.

arXiv comment22 pages, to appear Groups, Geometry, and Dynamics

References in corpus (1)

• Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms

Cited by in corpus (1)

• Cohomological finiteness conditions and centralisers in generalisations of Thompson's group V