Title:

Cohomological invariants for infinite groups

The main objects of interest in this thesis are H1Fgroups. These are groups that act on finitedimensional contractible CWspaces with finite stabilisers. Important examples of these are given by groups admitting a finitedimensional 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 H1Fgroup G admits a finitedimensional model for EFG. The natural choice for studying algebraically H1Fgroups is Fcohomology. This is a form of group cohomology relative to a Gset introduced by Nucinkis in 1999. In this theory there is a welldefined notion of Fcohomological dimension and we study its behaviour under taking group extensions. A conjecture of Nucinkis claims that every group G of finite Fcohomological dimension admits a finitedimensional 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 finitedimensional classifying space for proper actions is closed under taking extensions. In Chapter 3 we introduce and study the notion of Fhomological dimension and give an upper bound on the homological length of nonuniform 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 HFmembership 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 KrophollerMislin conjecture holds for a subclass of U and discuss its validity in general
