A Jorgensen-Thurston theorem for homomorphisms

TL;DR

This paper extends the Jorgensen-Thurston theorem to describe the structure of homomorphisms from finitely generated groups into torsion-free 3-dimensional Kleinian groups with bounded covolume, enriching understanding of their algebraic and geometric properties.

Contribution

It provides a new structural description of homomorphisms into torsion-free Kleinian groups, analogous to the classical Jorgensen-Thurston theorem in hyperbolic geometry.

Findings

Characterization of homomorphisms from finitely generated groups

Structural description for torsion-free Kleinian groups

Extension of Jorgensen-Thurston theorem to new setting

Abstract

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.