Second countable virtually free pro-p groups whose torsion elements have finite centralizer

Abstract

A second countable virtually free pro-$p$ group all of whose torsion elements have finite centralizer is the free pro-$p$ product of finite p-groups and a free pro-$p$ factor. The proof explores a connection between p-adic representations of finite p-groups and virtually free pro-p groups. In order to utilize this connection, we first prove a version of a remarkable theorem of A. Weiss for infinitely generated profinite modules that allows us to detect freeness of profinite modules. The proof now proceeds using techniques developed in the combinatorial theory of profinite groups. Using an HNN-extension, we embed our group into a semidirect product $F\rtimes K$ of a free pro-p group F and a finite p-group K that preserves the conditions on centralizers and such that every torsion element is conjugate to an element of K. We then show that the $\mathbb {Z}_pK$-module F / [F, F] is free using the detection theorem mentioned above. This allows us to deduce the result for $F\rtimes K$, and hence for our original group, using the pro-p version of the Kurosh subgroup theorem.