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

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 $\mathrm{F}_\infty$ and that this implies that also centralisers of finite subgroups are of type $\mathrm{F}_\infty$.