Bredon cohomological finiteness conditions for generalisations of Thompson 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 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-${\underline{\mathrm{F}}}_{\infty }$. 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-${\underline{\mathrm{F}}}_{\infty }$. We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.