Quasiconvexity and Dehn filling

TL;DR

This paper introduces a new condition on relatively hyperbolic Dehn fillings to control the behavior of relatively quasiconvex subgroups, leading to a new proof of virtual fibering of certain hyperbolic 3-manifolds.

Contribution

It presents a novel condition for Dehn fillings that manages relatively quasiconvex subgroups and extends previous results to the relative setting, offering new proofs and generalizations.

Findings

New condition on Dehn filling controlling subgroup behavior

Alternative proof of virtual fibering for hyperbolic 3-manifolds

Generalization of existing results to the relative setting

Abstract

We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic 3-manifolds, a result first proved by Wise. Additionally, we explain how the results of [2, Appendix A] can be generalized to the relative setting to control the relative height of relatively quasiconvex subgroups under appropriate Dehn fillings.