Absolute profinite rigidity, direct products, and finite presentability

Abstract

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form $\Gamma \times \Gamma$ where $\Gamma$ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if $P$ is a finitely generated, residually finite group with $\widehat{P}\cong\widehat{\Gamma\times\Gamma}$ then there is an embedding $P\hookrightarrow\Gamma\times\Gamma$ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for $P$.