# Enumerating limit groups

## TL;DR

This paper proves the set of limit groups is recursive, introduces algorithms for subgroup presentations, and calculates centralizers in relatively hyperbolic groups, advancing understanding of group theory structures.

## Contribution

It establishes the recursiveness of limit groups and provides algorithms for subgroup presentations and centralizer calculations in complex groups.

## Key findings

- The set of limit groups is recursive.
- Algorithms for finitely generated subgroup presentations are developed.
- Centralizers in relatively hyperbolic groups can be computed algorithmically.

## Abstract

We prove that the set of limit groups is recursive, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (a la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups.