#### Vol. 296, No. 2, 2018

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Presentations of generalisations of Thompson's group V

### Conchita Martínez-Pérez, Francesco Matucci and Brita Nucinkis

Vol. 296 (2018), No. 2, 371–403
##### 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.

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