Abstract. We show that Brin's generalisations 2V and 3V of the Thompson-Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

We consider the cohomology of Mackey functors for infinite groups and define the Mackeycohomological dimension cd M G of a group G. We will relate this dimension to other cohomological dimensions such as the Bredon-cohomological dimension cd F G and the relative cohomological dimension F-cdG. In particular, we show that for virtually torsion free groups the Mackeycohomological dimension is equal to both F-cdG and the virtual cohomological dimension.

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 a slightly weaker property, quasi-F ∞ , to that of a group possessing a finite type model for the classifying space for proper actions EG, and give criteria for the T versions of our groups to be of type quasi-F ∞ . We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.

Abstract. We show that the mapping class group of a closed surface admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension.

It is proved that every elementary amenable group of type FP ∞ admits a cocompact classifying space for proper actions.