The structure of limit groups over hyperbolic groups

TL;DR

This paper investigates the structure of limit groups over torsion-free hyperbolic groups, introducing model limit groups with favorable geometric properties and providing a framework for their algorithmic study.

Contribution

It defines model $ ext{ extGamma}$-limit groups with good geometric features and establishes a canonical resolution framework for analyzing all homomorphisms from limit groups to hyperbolic groups.

Findings

Model $ ext{ extGamma}$-limit groups are always relatively hyperbolic.

A canonical resolution process encodes all homomorphisms factoring through given resolutions.

The framework enables enumeration and algorithmic study of $ ext{ extGamma}$-limit groups.

Abstract

Let $\Gamma$ be a torsion-free hyperbolic group. We study $\Gamma$--limit groups which, unlike the fundamental case in which $\Gamma$ is free, may not be finitely presentable or geometrically tractable. We define model $\Gamma$--limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary $\Gamma$--limit group $L$, we canonically construct a strict resolution of a model $\Gamma$--limit group, which encodes all homomorphisms $L\to \Gamma$ that factor through the given resolution. We propose this as the correct framework in which to study $\Gamma$--limit groups algorithmically. We enumerate all $\Gamma$--limit groups in this framework.