Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group <i>V</i>

Abstract

Abstract We consider generalisations of Thompson’s group V , denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>Σ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${V_{r}(\Sigma)}$ , which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>Σ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mo>F</m:mo> <m:mi>∞</m:mi> </m:msub> </m:math> ${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mo>F</m:mo> <m:mi>∞</m:mi> </m:msub> </m:math> ${\operatorname{F}_{\infty}}$ .