On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups

We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.Comment: 15 pages; revised versio

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

We consider generalisations of Thompson's group $V$, denoted $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, $V_r(\Sigma)$ is the full automorphism group of a Cantor-algebra. Under some further minor restrictions, we prove that these groups are of type $\mathrm{F}_\infty$ and that this implies that also centralisers of finite subgroups are of type $\mathrm{F}_\infty$.Comment: 19 pages, 2 figures. Revised version. The original submission has now been split into two papers. The current submission contains the first one. The second part is being reworked and will be reposted soon independently. Lemma 4.8 was incorrect as stated and has since been rectified. The results of the paper are unchange

Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

We study the groupQV , the self-maps of the infinite 2-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups QF , QT , zQ T , and zQV , prove that QF , zQ T , and zQV are of type F1, and calculate finite presentations for them. We calculate the normal subgroup structure of all 5 groups, the Bieri-Neumann-Strebel-Renz invariants of QF , and discuss the relationship of all 5 groups with other generalisations of Thompson's groups.</p

Irrational-slope versions of thompson's groups T and V

In this paper, we consider the T - and V -versions, Tt and Vt , of the irrational slope Thompson group Ft considered in J. Burillo, B. Nucinkis and L. Reeves [An irrational-slope Thompson's group, Publ. Mat. 65 (2021), 809–839]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for T and V . We also show that Tt and Vt have index- 2 normal subgroups, unlike their original Thompson counterparts T and V . These index- 2 subgroups are shown to be simple.Peer ReviewedPostprint (published version

An irrational-slope Thompson’s group

The purpose of this paper is to study the properties of the irrational-slope Thompson’s group Ft introduced by Cleary in [11]. We construct presentations, both finite and infinite, and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson’s group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of F in Ft are undistorted.Peer ReviewedPostprint (published version

Cohomological finiteness conditions for elementary amenable groups

It is proved that every elementary amenable group of type $FP_\infty$ admits a cocompact classifying space for proper actions