The profinite completion of accessible groups

Abstract

A finitely generated residually finite group G is an OE-group if any action of its profinite completion G on a profinite tree with finite edge stabilizers admits a global fixed point. In this paper, we study the profinite genus of free products G 1 * H G 2 of OE-groups G 1 , G 2 with finite amalgamation H. Given such G 1 , G 2 , H we give precise formulas for the number of isomorphism classes of G 1 * H G 2 and of its profinite completion. We compute the genus of G 1 * H G 2 and list various situations when the formula for the genus simplifies.