Profinite Subgroup Accessibility and Recognition of Amalgamated Factors

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.