Profinite Subgroup Accessibility and Recognition of Amalgamated Factors

TL;DR

This paper explores the structure of accessible subgroups in profinite groups, establishing new links between their algebraic properties and splittings, and extending results on free factors in amalgamated products.

Contribution

It introduces a novel characterization of accessible subgroups via continuous derivations and extends free factor recognition results to profinite and abstract groups.

Findings

Accessible subgroups are kernels of continuous derivations.

Splittings of abstract groups can be deduced from their profinite completions.

Finitely generated subgroups in profinite amalgams are factors in certain group decompositions.

Abstract

We investigate accessible subgroups of a profinite group $G$, i.e. subgroups $H$ appearing as vertex groups in a graph of profinite groups decomposition of $G$ with finite edge groups. We prove that any accessible subgroup $H \leq G$ arises as the kernel of a continuous derivation of $G$ in a free module over its completed group algebra. This allows us to deduce splittings of an abstract group from splittings of its profinite completion. We prove that any finitely generated subgroup $\Delta$ of a finitely generated virtually free group $\Gamma$ whose closure is a factor in a profinite amalgamated product $\widehat{\Gamma} = \overline{\Delta} \amalg_K L$ along a finite $K$ must be a factor in an amalgamated product $\Gamma = \Delta \ast_\chi \Lambda$ along some $\chi \cong K$. This extends previous results of Parzanchevski--Puder, Wilton and Garrido--Jaikin-Zapirain on free factors.