The structure of limit groups over hyperbolic groups

Abstract

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