Presentations of generalisations of Thompson's group V

Conchita Martínez-Pérez, Francesco Matucci and Brita Nucinkis

Vol. 296 (2018), No. 2, 371–403

DOI: 10.2140/pjm.2018.296.371

Abstract

We consider generalisations of Thompson’s group $V$, denoted by $V_r\left(\Sigma \right)$, which also include the groups of Higman, Stein and Brin. We showed earlier (Forum Math. 28:5 (2016), 909–921) that under some mild conditions these groups and centralisers of their finite subgroups are of type $F_{\infty }$. Under more general conditions we show that the groups $V_r\left(\Sigma \right)$ are finitely generated and, under the mild conditions mentioned above for which they are of type $F_{\infty }$ and hence finitely presented, we give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then show how to apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure.

Keywords

generalized Thompson groups, finite presentations

Mathematical Subject Classification 2010

Primary: 20J05

Milestones

Received: 17 January 2017

Revised: 12 December 2017

Accepted: 12 December 2017

Published: 16 July 2018