A Jørgensen–Thurston theorem for homomorphisms

Abstract

In this note, we provide a description of the structure homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jørgensen-Thurston Theorem in hyperbolic geometry. YI LIUfinitely-generated case can be reduced to the finitely-presented case, (Proposition 4.12).Acknowledgement. The author thanks Ian Agol and Daniel Groves for helpful conversations. PreliminariesIn this section, we recall some notions and results related to the topic of this paper.2.1. The Jørgensen-Thurston Theorem. For convenience of our discussion, we adopt the following notations and state the results in terms of Kleinian groups. Cf. [Th] for facts mentioned in this subsection.Let Γ be a torsion-free Kleinian group of finite covolume, namely, which has a fundamental domain in H 3 of finite volume. Then Γ has at most finitely many of conjugacy-classes parabolic subgroups, represented by subgroups P 1 , • • • , P q , (q ≥ 0), each isomorphic to a rank-2 free abelian group. We often call these chosen subgroups cusp representatives of Γ. By a slope-tupleor ambiguously, w.r.t. Γ), we shall mean that for each 1 ≤ j ≤ q, the j-th component ζ j is either trivial or a primitive element in P j . For any slopetuple ζ of Γ, we denote the Dehn filling of Γ along ζ as:i.e. Γ quotienting out the normal closure of the ζ j 's, and often denote the quotient epimorphism as:By Thurston's Hyperbolic Dehn-Filling Theorem ([Th, Theorem 5.8.2]), for generic slope-tuples ζ (avoiding finitely many primitive choices for each component), Γ ζ is isomorphic to a torision-free Kleinian group of finite covolume, indeed, strictly less than that of Γ if ζ is nonempty. These Γ ζ 's are usually called hyperbolic Dehn fillings of Γ. In fact, one may choose faithful Kleinian representations ρ ζ : Γ ζ → PSL 2 (C), so that for any sequence {ζ n } of slope-tuples, there is a subsequence for which the induced representations {ρ ζn • ι ζn } of Γ strongly converges. Moreover, if for each j-th component, {ζ j n } has no bounded subsequence of primitive elements in P j , then {ρ ζn • ι ζn } strongly converges to the inclusion Γ ⊂ PSL 2 (C).The following rephrases [Th, Theorem 5.12.1]:Theorem 2.1 (Jørgensen-Thurston). For any V > 0, there exist finitely many torsion-free Kleinian groups Γ 1 , • • • , Γ k of covolume at most V , (together with chosen cusp representatives), such that any torsion-free Kleinian group of covolume at most V is isomorphic to the hyperbolic Dehn filling (Γ i ) ζ of some Γ i along some slope-tuple ζ of Γ i .Remark 2.2. It is also implied from the proof that hyperbolic Dehn fillings decreases the covolume.

Discrete representations of finitely generated groups into PSL(2,R)${\rm PSL}(2,\mathbb {R})$

“…Factorization theorems of the same flavor have been proved by Agol and Liu in [1] for maps from a finitely presented group $$G$$ to the fundamental group of an orientable aspherical compact 3‐manifold $$M$$. (As an application of their factorization theorems, Liu proves a factorization theorem similar to Theorem 1.1 for torsion‐free Kleinian groups of uniformly bounded covolume in [23]. ) In their theorems, instead of a cone point with sufficiently large order being replaced by a puncture, a sufficiently short simple closed geodesic in a hyperbolic piece (or an exceptional fiber at a sufficiently sharp cone point in a Seifert fibered piece) is drilled out.…”

“…Agol and Liu's proof is topological and does not seem to generalize easily to the case where $$G$$ is finitely generated. (However, when the 3‐manifold $$M$$ is hyperbolic, Liu did generalized their theorem to finitely generated case in [23]. ) Our proof of the factorization theorem relies on the theory of group actions on $$\mathbb {R}$$‐trees and Sela's theory of limit groups.…”