Cohomology and profinite topologies for solvable groups of finite rank

Abstract

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^{\infty }}$. We show that if $G$ is nilpotent, then the pro-$p$ completion map $G\to {\hat{G}}_p$ induces an isomorphism $H^{\ast }({\hat{G}}_p,M)\to H^{\ast }(G,M)$ for any discrete ${\hat{G}}_p$-module $M$ of finite $p$-power order. For the general case, we prove that $G$ contains a normal subgroup $N$ of finite index such that the map $H^{\ast }({\hat{N}}_p,M)\to H^{\ast }(N,M)$ is an isomorphism for any discrete ${\hat{N}}_p$-module $M$ of finite $p$-power order. Moreover, if $G$ lacks any $C_{p^{\infty }}$-sections, the subgroup $N$ enjoys some additional special properties with respect to its pro-$p$ topology.

arXiv commentThis paper supersedes arXiv:1009.2645v5: the two theorems in the introduction to the latter paper are both corollaries to Theorem 1.1 in the present paper. In the second version, Theorem 1.1 is expressed in a slightly more general form than in the first version