Finiteness conditions in group cohomology.
PhD thesis, University of Glasgow.
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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 LHF-group, then the set F(G) is either
finite or cofinite. Kropholler's theorem does not, however, give a characterisation of the LHF-groups 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 LHF-groups 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 (polycyclic-by-finite) 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 non-trivial 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 (polycyclic-by-finite) 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 LHF-groups with cohomology almost everywhere finitary. In particular, we show that if G is an LHF-group with cohomology almost everywhere finitary, then GxZ has an Eilenberg-Mac Lane space K(GxZ,1) with finitely many n-cells 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.
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