Cohomology and profinite topologies for solvable groups of finite rank

TL;DR

This paper investigates the cohomological properties of solvable groups of finite rank with certain restrictions, establishing isomorphisms between group cohomology and pro-p completions, especially for nilpotent groups and groups without specific infinite cyclic subgroups.

Contribution

It proves new isomorphism results between group cohomology and pro-p completions for solvable groups under specific conditions, extending known results to broader classes.

Findings

For nilpotent groups, the pro-p completion induces cohomology isomorphisms.

Existence of a finite index subgroup with similar cohomological properties.

Additional properties when the group lacks $C_{p^inite}$-sections.

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.