Subgroups of genus-2 quasi-Fuchsian groups and cocompact Kleinian groups

Rao, Zhenghao

doi:10.2140/gt.2025.29.495

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Subgroups of genus-$2$ quasi-Fuchsian groups and cocompact Kleinian groups

Zhenghao Rao

Geometry & Topology 29 (2025) 495–548

DOI: 10.2140/gt.2025.29.495

Abstract

We wish to control the geometry of some surface subgroups of a cocompact Kleinian group. More precisely, provided any genus-2 quasi-Fuchsian group $Γ$ and cocompact Kleinian group $G$ , then for any $ν > 0$ we will find a surface subgroup $H$ of $G$ that is $(1 + ν)$ -quasiconformally conjugate to a finite-index subgroup $F < Γ$ .

Keywords

quasi-Fuchsian groups, hyperbolic $3$-manifolds, surface subgroups

Mathematical Subject Classification

Primary: 22E40, 30C62, 30F40, 57K32

Secondary: 37A25

References

Publication

Received: 22 March 2023

Revised: 5 November 2023

Accepted: 17 February 2024

Published: 4 March 2025

Proposed: Benson Farb

Seconded: David Fisher, Mladen Bestvina

Authors

Zhenghao Rao
Department of Mathematics Brown University Providence, RI United States	Department of Mathematics Rutgers University Piscataway, NJ United States

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Volume 29, issue 1 (2025)