Title:

Finiteness conditions in group cohomology

In this thesis we investigate groups whose nth cohomology functors commute with filtered colimits for all sufficiently large n. In Chapter 1 we introduce some basic definitions and important background material. We make the definition that a group G has cohomology almost everywhere finitary if and only if the set F(G) of natural numbers n for which the nth cohomology of G commutes with filtered colimits is cofinite. We also introduce Kropholler's class LHF of locally hierarchically decomposable groups. We then state a key result of Kropholler, which establishes a dichotomy for this class: If G is an LHFgroup, then the set F(G) is either finite or cofinite. Kropholler's theorem does not, however, give a characterisation of the LHFgroups with cohomology almost everywhere finitary, and this is precisely the problem that we are interested in. In Chapter 2 we investigate algebraic characterisations of certain classes of LHFgroups with cohomology almost everywhere finitary. In particular, we establish sufficient conditions for a group in the class H1F to have cohomology almost everywhere finitary. We prove a stronger result for the class of groups of finite virtual cohomological dimension over a ring R of prime characteristic p, and use this result to answer an open question of Leary and Nucinkis. We also consider the class of locally (polycyclicbyfinite) groups, and show that such a group G has cohomology almost everywhere finitary if and only if G has finite virtual cohomological dimension and the normalizer of every nontrivial finite subgroup of G is finitely generated. We then change direction in Chapter 3, and show an interesting connection between this problem and the problem of group actions on spheres. In particular, we show that if G is an infinitely generated locally (polycyclicbyfinite) group with cohomology almost everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere. Finally, in Chapter 4 we provide a topological characterisation of the LHFgroups with cohomology almost everywhere finitary. In particular, we show that if G is an LHFgroup with cohomology almost everywhere finitary, then GxZ has an EilenbergMac Lane space K(GxZ,1) with finitely many ncells for all sufficiently large n. It is an open question as to whether the LHF restriction can be dropped here. We also show that the converse statement holds for arbitrary G.
