A recipe for constructing non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups

TL;DR

This paper introduces a general method to construct non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups, expanding understanding of group properties and providing explicit examples.

Contribution

It presents a novel construction technique using the image extension theorem to generate such groups, including two concrete examples.

Findings

Established a general construction method for these groups

Provided explicit examples demonstrating the method

Extended knowledge on the properties of relatively hyperbolic groups

Abstract

Very recently, Kim and Lee presented an example of a non-Hopfian relatively hyperbolic group with a Hopfian peripheral subgroup, demonstrating a counterexample to Osin's well-known question (Problem 5.5). In this paper, we provide a general construction method using the so-called image extension theorem to generate non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups. Additionally, we provide two specific examples using this construction method.