Cohomological finiteness conditions for elementary amenable groups

It is proved that every elementary amenable group of type $FP_\infty$ admits a cocompact classifying space for proper actions

On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups

We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy

Is there an algebraic characterisation for finite-dimensional proper G-spaces?

We shall study properties of groups having finite cohomological dimension relative to the family of all finite subgroups. We also compare these groups with those satisfying various suggested algebraic analogues to group-actions on finite dimensional proper G-spaces

On dimensions in Bredon homology

We define a homological and cohomological dimension of groups in the context of Bredon homology and compare the two quantities. We apply this to describe the Bredon-homological dimension of nilpotent groups in terms of the Hirsch-rank. In particular this implies that for virtually torsion-free nilpotent groups the Bredon cohomological dimension is equal to the virtual cohomological dimension

Cohomological finiteness properties of the Brin-Thompson-higman groups 2V and 3V

We show that Brin's generalisations 2V and 3V of the Thompson-Higman group V are of type FP_\infty. Our methods also give a new proof that both groups are finitely presented

Some groups of type VF

A group is of type VF if it contains a finite-index subgroup which has a finite classifying space. We construct groups of type VF in which the centralizers of some elements of finite order are not of type VF and groups of type VF containing infinitely many conjugacy classes of finite subgroups. From these examples it follows that a group G of type VF need not admit a finite-type or finite classifying space for proper actions (sometimes also called the universal proper G-space). We construct groups G for which the minimal dimension of a universal proper G-space is strictly greater than the virtual cohomological dimension of G. Each of our groups embeds in some general linear group over the rational integers. Applications to algebraic K-theory of group algebras and topological K-theory of group C*-algebras are also considered. The groups are constructed as finite extensions of Bestvina-Brady groups

Cohomological dimension of Mackey functors for infinite groups

We consider the cohomology of Mackey functors for infinite groups and define the Mackey-cohomological dimension of a group G. We relate this dimension to other cohomological dimensions such as the Bredon cohomological dimension and the relative cohomological dimension. In particular we show that for virtually torsion free groups the Mackey cohomological dimension is equal to both the relative cohomological dimension and the virtual cohomological dimension.<br/

Periodic cohomology and subgroups with bounded Bredon cohomological dimension

Mislin and Talelli showed that a torsion-free group in Kropholler's class &lt;b&gt;H&lt;/b&gt;F with periodic cohomology after some steps has finite cohomological dimension. In this note we look at similar questions for groups with torsion by considering Bredon cohomology. In particular we show that every elementary amenable group acting freely and properly on some &lt;b&gt;R&lt;/b&gt;^n x S^m admits a finite dimensional classifying space for proper actions.<br/

On Bredon homology for elementary amenable groups

We show that for elementary amenable groups the Hirsch length is equal to the Bredon homological dimension. This also implies that countable elementary amenable groups admit a finite-dimensional model for $\underbar{EG}$ of dimension less than or equal to the Hirsch length plus one. Some remarks on groups of type $FP_\infty$ are also made.<br/

Bounding the orders of finite subgroups

Suppose that G is a group of rational cohomological dimension n and that G is of type FP(n) over the integers. Under these hypotheses we show that there is a bound on the orders of finite subgroups of G. This extends a result of P. H. Kropholler, who obtained the same conclusion for G of finite rational cohomological dimension and of type FP(infinity) over the integers. For each n, there are groups G of type FP(n-1) over the integers and of rational cohomological dimension n for which there is no bound on the orders of finite subgroups