Cohomology of Profinite Groups

The aim of this thesis is to study profinite groups of type FPn. These are groups G which admit a projective resolution P of Ẑ as a ẐJGK-module such that P0, . . . , Pn are finitely generated, so this property can be studied using the tools of profinite group cohomology. In studying profinite groups it is often useful to consider their cohomology groups with profinite coefficients, but pre-existing theories of profinite cohomology do not allow profinite coefficients in sufficient generality for our purposes. Therefore we develop a new framework in which to study the homology and cohomology of profinite groups, which allows second-countable profinite coefficients for all profinite groups. We prove that many of the results of abstract group cohomology hold here, including Shapiro’s Lemma, the Universal Coefficient Theorem and the Lyndon-Hochschild-Serre spectral sequence. We then use these homology and cohomology theories to study how being of type FPn controls the structure of a profinite group, and vice versa. We show for all n that the class of groups of type FPn is closed under extensions, quotients by subgroups of type FPn, proper amalgamated free products and proper HNNextensions, and hence that elementary amenable profinite groups of finite rank are of type FP∞. We construct profinite groups of type FPn but not FPn+1 for all n. Finally, we develop the theory of signed profinite permutation modules, and use these as coefficients for group cohomology to show that torsion-free soluble pro-p groups of type FP∞ have finite rank.