Absolute profinite rigidity, direct products, and finite presentability

TL;DR

This paper demonstrates the existence of finitely presented, residually finite groups that are uniquely determined by their finite quotients within finitely presented groups but not within all finitely generated groups, highlighting nuanced rigidity properties.

Contribution

It introduces new examples of profinitely rigid groups formed as direct products of 3-manifold groups and analyzes their embedding and isomorphism properties.

Findings

Existence of finitely presented, residually finite groups with unique profinite completions in certain classes.

Construction of groups with infinitely many non-isomorphic groups sharing the same profinite completion.

Identification of conditions under which these groups embed into their profinite completions.

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 $P$.