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

Forum Mathematicum , 28(5) (2016), 909-921
http://arxiv.org/abs/1309.7858 and https://doi.org/10.1515/forum-2014-0176

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 F_\infty. We show that, provided that V_r(\Sigma) is of type F_\infty, centralisers of finite subgroups are also of type F_\infty, and give an explicit finite presentation of these centralisers in this case. Upon weakening the hypotheses on the Cantor-algebra, it can be shown that the groups V_r(\Sigma) are still finitely generated.