Presentations of Generalisations of Thompson’s Group V

We consider generalisations of Thompson’s group V , denoted by Vr(Σ), which also include the groups of Higman, Stein and Brin. It was shown in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type F∞. Under more general conditions we show that the groups Vr(Σ) are finitely generated and, under the mild conditions mentioned above for which they are of type F∞ 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.