# Immutability is not uniformly decidable in hyperbolic groups

## TL;DR

This paper proves that there is no general algorithm to determine whether a finitely generated subgroup of a torsion-free hyperbolic group is immutable, highlighting fundamental limits in algorithmic group theory.

## Contribution

It establishes the non-existence of a uniform algorithm for recognizing immutability in hyperbolic groups, answering a previously open question.

## Key findings

- No uniform algorithm exists for recognizing immutability.
- Immutability is not a decidable property in this context.
- The result impacts understanding of subgroup properties in hyperbolic groups.

## Abstract

A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability, answering a uniform version of a question asked by the authors.