Infinitely Generated virtually free pro-$p$ groups and $p$-adic representations

TL;DR

This paper extends classical results about virtually free groups to the pro-$p$ setting, showing how such groups act on trees and embed into free pro-$p$ products using $p$-adic representation theory.

Contribution

It proves a pro-$p$ analogue of a classical theorem about virtually free groups acting on trees, employing $p$-adic representation theory instead of Stallings' ends theory.

Findings

Virtually free pro-$p$ groups act on trees with finite vertex stabilizers.

Under certain conditions, such groups embed into free pro-$p$ products with finite $p$-groups.

The proof introduces $p$-adic representation theory as a tool in the pro-$p$ context.

Abstract

We prove the pro-$p$ version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro-$p$ group $G$ has finite centralizes of all non-trivial torsion elements more stronger statement is proved: $G$ embeds into a free pro-$p$ product of a free pro-$p$ group and finite $p$-group. The integral $p$-adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro-$p$ case.