The structure of some virtually free pro-$p$ groups

Abstract:

We prove two conjectures on pro-$p$ groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro-$p$ group which has an open free pro-$p$ subgroup of index $p$ is a free pro-$p$ product $H_0*(S_1\times H_1)*\cdots *(S_m\times H_m)$, where the $H_i$ are free pro-$p$ of finite rank and the $S_i$ are cyclic of order $p$. The second says that if $F$ is a free pro-$p$ group of finite rank and $S$ is a finite $p$-group of automorphisms of $F$, then $\operatorname {Fix}(S)$ is a free factor of $F$. The proofs use cohomology, and in particular a “Brown theorem” for profinite groups.