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 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.

Hierarchically cocompact classifying spaces for mapping class groups of surfaces

“…In the authors introduced a similar condition for the family of finite subgroups, quasi‐, which asks for a group to have, for any , finitely many conjugacy classes of finite subgroups of order , and that normalisers of all finite subgroups are of type . [, Theorem 4.9] shows that generalised Thompson groups, which are automorphism groups of valid, bounded, and complete Cantor‐algebras are quasi‐ and hence are .…”

“…By the universal property for classifying spaces for families, this is equivalent to saying that there is a model for which is a mapping telescope of cocompact (finite type) models for . This follows from an argument analogous to [, Theorem 6.11].…”

“…In particular, this answers a question of Lück for mapping class groups of surfaces.Definition 1.1. We say a G-CW-complex X is a hierarchically cocompact (hierarchically finite type) model for E F G if there are nested families F i (i ∈ N) such that F = i∈N F i , and that there are cocompact (finite type) models for E Fi G for all i ∈ N.By the universal property for classifying spaces for families, this is equivalent to saying that there is a model for E F G which is a mapping telescope of cocompact (finite type) models for E Fi G. This follows from an argument analogous to [18, Theorem 6.11].Note that under the additional hypothesis that for a countable collection of families {F i } (i∈N) that there is cocompact (finite type) model for E Fi∩Fj G for all i, j ∈ N, then we can always ensure that we have a nested family as above, see the construction in the first paragraph of the proof of Lemma 2.6 below. Definition 1.1 was motivated by the following conjecture of Juan-Pineda and Leary.…”

The infinite simple group $V$ of Richard J. Thompson: presentations by permutations

“…When n is odd, one must pass to the commutator subgroup of index 2, reflecting the observation that the corresponding split relations in G n,r do not change the parity of any decomposition as a product of transpositions.) Other families include the Brin-Thompson groups nV for which V = 1V , see [6], and the groups nV m,r that generalise the previous two families, see [20], and where we have similar simplicity considerations, see [7]. The finite presentability of these groups comes from the much stronger fact that they are all in fact F ∞ groups.…”

On Thompson's group T and algebraic K-theory

“…We proved Theorem 1.3 in 2007 and only lately became aware that essentially the same result (but without the observation about normalizers, which is important for our application) appeared in Matucci's 2008 thesis [Mat08, Theorem 7.1.5], and was subsequently generalized in [MPN13,MPMN16]. The full details of our proof of Theorem 1.3 are given in Section 2.…”