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We consider generalisations of Thompson's group V, denoted by Vr(Σ), which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, Vr(Σ) is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type F∞ and that this implies that also centralisers of finite subgroups are of type F∞. [ABSTRACT FROM AUTHOR]

Kropholler and Mislin conjectured that groups acting admissibly on a finite-dimensional G-CW-complex with finite stabilisers admit a finite-dimensional model for E픉 G, the classifying space for proper actions. This conjecture is known to hold for groups with bounded torsion. In this note we consider a large class of groups 픘 containing the above and many known examples with unbounded torsion. We show that the conjecture holds for a large subclass of 픘. [ABSTRACT FROM AUTHOR]

We show that for soluble groups of type FP n, centralisers of finite subgroups need not be of type FP n. [ABSTRACT FROM AUTHOR]

Let  denote the class of finite groups, and let  denote the subclass consisting of groups of prime-power order. We study group actions on topological spaces in which all stabilizers lie either in  or in . We compare the classifying spaces for actions with stabilizers in  and , the Kropholler hierarchies built on  and , and group cohomology relative to  and to . In terms of standard notation, we show that  ⊂ 1 ⊂ 1, with all inclusions proper; that  = ; that  H*( G; –); =  H*( G; –); and that E G is finite-dimensional if and only if E G is finite-dimensional and every finite subgroup of G is in . [ABSTRACT FROM AUTHOR]

We show that for soluble groups of type FP n, centralisers of finite subgroups need not be of type FP n. [ABSTRACT FROM AUTHOR]