Presentations of generalisations of Thompson's group $V$

We consider generalisations of Thompson’s group $V$, denoted by $V_r(\mathrm{\Sigma })$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type ${\mathrm{F}}_{\mathrm{\infty }}$. Under more general conditions we show that the groups $V_r(\mathrm{\Sigma })$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure.