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{\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.