Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

TL;DR

This paper proves that every finite covolume Kleinian group contains a nearly Fuchsian surface subgroup, constructed via quasiconformal conjugation, extending the understanding of subgroup structures in Kleinian groups.

Contribution

It establishes the existence of nearly Fuchsian surface subgroups in all finite covolume Kleinian groups for any K > 1, generalizing previous results.

Findings

Existence of K-quasiconformally conjugate surface subgroups

Extension of Kahn and Markovic's work to all finite covolume groups

Construction of nearly Fuchsian surface subgroups

Abstract

Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.