# Hyperbolic groups that are not commensurably coHopfian

## TL;DR

This paper demonstrates that certain torsion-free one-ended hyperbolic groups, including fundamental groups of simple surface amalgams, are not commensurably coHopfian, contrasting with Sela's earlier results.

## Contribution

It introduces examples of hyperbolic groups that are not commensurably coHopfian, expanding understanding of subgroup properties in hyperbolic groups.

## Key findings

- Existence of torsion-free one-ended hyperbolic groups that are not commensurably coHopfian
- Fundamental groups of simple surface amalgams are not commensurably coHopfian
- Contrasts with Sela's result on coHopfian hyperbolic groups

## Abstract

Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.