Profinite subgroup accessibility and recognition of amalgamated factors

Abstract

We investigate accessible subgroups of a profinite group G , that is, 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 \hat{\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.