Dehn filling in semisimple Lie groups

TL;DR

This paper extends Thurston's hyperbolic Dehn filling theorem to higher-rank semisimple Lie groups, providing criteria for deformations of certain subgroups to remain geometrically finite and ensuring continuous variation of limit sets.

Contribution

It generalizes Dehn filling concepts to semisimple Lie groups and establishes conditions under which deformations preserve geometric finiteness and Anosov properties.

Findings

Deformations of extended geometrically finite subgroups remain geometrically finite.

Deformations of relatively Anosov subgroups can be non-relatively Anosov under certain conditions.

Limit sets vary continuously during these deformations.

Abstract

We generalize one part of Thurston's hyperbolic Dehn filling theorem to arbitrary-rank semisimple Lie groups by showing that certain deformations of extended geometrically finite subgroups of a semisimple Lie group are still extended geometrically finite. As a special case, our theorem gives a criterion which guarantees that a deformation of a relatively Anosov subgroup is (non-relatively) Anosov, and also ensures that limit sets vary continuously. Our result also applies to several higher-rank examples in convex projective geometry which are outside of the relatively Anosov setting.