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

Abstract

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 ${\mathrm{F}}_{\infty }$.

19 pages, 2 figures. Revised version. The original submission has now been split into two papers. The current submission contains the first one. The second part is being reworked and will be reposted soon independently. Lemma 4.8 was incorrect as stated and has since been rectified. The results of the paper are unchanged

References in corpus (1)

• On Thompson's group T and algebraic K-theory

Cited by in corpus (1)

• Hierarchically cocompact classifying spaces for mapping class groups of surfaces