The strong profinite genus of a finitely presented group can be infinite

Abstract

We construct the first examples of finitely-presented, residually-finite groups $\Gamma$ that contain an infinite sequence of non-isomorphic finitely-presented subgroups $P_n\hookrightarrow \Gamma$ such that the inclusion maps induce isomorphisms of profinite completions ${\hat{P}}_n\cong \hat{\Gamma }$.