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Co-Hopfianity is not a profinite property

Hyungryul Baik, Wonyong Jang

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.

Co-Hopfianity is not a profinite property

Abstract

We exhibit two finitely generated residually finite groups and with isomorphic profinite completions , such that is co-Hopfian while is not. The construction utilizes Wise's residually finite version of the Rips construction applied to a finitely presented acyclic group with trivial profinite completion and a strong universality property. A key feature of our approach is the construction of as a preimage subgroup of which is conjugate to a proper subgroup of itself. This renders the non-co-Hopfianity of immediate without requiring a detailed structural analysis of the Rips kernel.
Paper Structure (11 sections, 13 theorems, 24 equations)

This paper contains 11 sections, 13 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hopfian and co-Hopfian groups
  4. Profinite completion
  5. Wise's residually finite Rips construction
  6. A universal acyclic group
  7. The proof of the main result
  8. The construction of a co-Hopfian group $G$
  9. A self-embedding via conjugation inside $U$
  10. Constructing $H$ as a preimage subgroup of $G$
  11. Profinite completions and finiteness properties

Key Result

Theorem 1.1

There exist finitely generated residually finite groups $G$ and $H$ such that:

Theorems & Definitions (26)

  • Theorem 1.1: Theorem \ref{['thm:main']}
  • Theorem 2.1: sela1997structure
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: bridson2010schur
  • Remark 2.6
  • Theorem 2.7: wise2003residually
  • Lemma 2.8: bridson2019homology
  • Proposition 2.9
  • ...and 16 more