Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups

TL;DR

This paper investigates the geometric relationships between surface subgroups of cocompact Kleinian groups and genus-2 quasi-Fuchsian groups, establishing conditions for quasiconformal conjugacy and subgroup control.

Contribution

It introduces a method to find surface subgroups of cocompact Kleinian groups that are quasiconformally conjugate to subgroups of genus-2 quasi-Fuchsian groups, enhancing understanding of their geometric structure.

Findings

Existence of surface subgroups with controlled quasiconformal conjugacy.

Explicit construction of subgroups with bounded quasiconformal distortion.

Extension of subgroup control techniques to genus-2 quasi-Fuchsian groups.

Abstract

In this paper, we want to control the geometry of some surface subgroups of a cocompact Kleinian group. More precisely, provided any genus-2 quasi-Fuchsian group $\Gamma$ and cocompact Kleinian group $G$, then for any $K>1$, we will find a surface subgroup $H$ of $G$ that is $K$-quasiconformally conjugate to a finite index subgroup $F<\Gamma$.