Co-Hopfianity is not a profinite property

TL;DR

This paper demonstrates that co-Hopfianity, a property of groups, is not determined solely by their profinite completions by constructing examples of finitely generated groups with identical profinite completions but differing co-Hopfian properties.

Contribution

The authors construct explicit examples of finitely generated residually finite groups with isomorphic profinite completions where one is co-Hopfian and the other is not, showing co-Hopfianity is not a profinite property.

Findings

Constructed groups with identical profinite completions but different co-Hopfianity.

Used Wise's residually finite Rips construction with a group having trivial profinite completion.

Established non-co-Hopfianity through subgroup conjugation without detailed Rips kernel analysis.

Abstract

We exhibit two finitely generated residually finite groups $G$ and $H$ with isomorphic profinite completions $\widehat{G} \cong \widehat{H}$, such that $G$ is co-Hopfian while $H$ is not. The construction utilizes Wise's residually finite version of the Rips construction applied to a finitely presented acyclic group $U$ with trivial profinite completion and a strong universality property. A key feature of our approach is the construction of $H$ as a preimage subgroup of $G$ which is conjugate to a proper subgroup of itself. This renders the non-co-Hopfianity of $H$ immediate without requiring a detailed structural analysis of the Rips kernel.