Cohomological invariants for infinite groups

Abstract

The main objects of interest in this thesis are H1F-groups. These are groups that act on finite-dimensional contractible CW-spaces with finite stabilisers. Important examples of these are given by groups admitting a finite-dimensional classifying space for proper actions EFG. A large part of the thesis is motivated by an old conjecture of Kropholler and Mislin claiming that every H1F-group G admits a finite-dimensional model for EFG. The natural choice for studying algebraically H1F-groups is F-cohomology. This is a form of group cohomology relative to a G-set introduced by Nucinkis in 1999. In this theory there is a well-defined notion of F-cohomological dimension and we study its behaviour under taking group extensions. A conjecture of Nucinkis claims that every group G of finite F-cohomological dimension admits a finite-dimensional model for EFG. Note that it is unknown whether the class H1F is closed under taking extensions. It is also unknown whether the class of groups admitting a finite-dimensional classifying space for proper actions is closed under taking extensions. In Chapter 3 we introduce and study the notion of F-homological dimension and give an upper bound on the homological length of non-uniform lattices on locally finite CATp0q polyhedral complexes, giving an easier proof that generalises an important result for arithmetic groups over function fields, due to Bux and Wortman. The first Grigorchuk group G was introduced in 1980 and has been extensively studied since due to its extraordinary properties. The class HF of hierarchically decomposable groups was introduced by Kropholler in 1993. There are very few known examples of groups that lie outside HF. We answer the question regarding the HF-membership of G by showing that G lies outside HF. In the final chapter we introduce a new class of groups U, and show that the Kropholler-Mislin conjecture holds for a subclass of U and discuss its validity in general

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