# Virtual retraction and Howson's theorem in pro-$p$ groups

## TL;DR

This paper investigates properties of Demushkin groups, proving a virtual retraction property, a version of Howson's theorem, and confirming conjectures related to pro-$p$ M. Hall groups, with implications for Galois groups.

## Contribution

It establishes a virtual retraction property for finitely generated subgroups, proves Howson's theorem for Demushkin groups, and confirms two conjectures on pro-$p$ M. Hall groups.

## Key findings

- Finitely generated subgroups have open supergroups with retraction homomorphisms.
- Intersections of finitely generated subgroups are finitely generated with explicit bounds.
- Properties are preserved under free pro-$p$ products, extending to Galois groups.

## Abstract

We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $\tau \colon U \to K$ satisfying $\tau(k) = k$ for every $k \in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products, and deduce that Howson's theorem holds for the Sylow subgroups of the absolute Galois group of a number field. Finally, we confirm two conjectures of Ribes, thus classifying the finitely generated pro-$p$ M. Hall groups.