Dynamics for discrete subgroups of $\operatorname{SL}_2(\mathbb C)$

TL;DR

This paper discusses recent advances in the theory of Kleinian groups and 3-dimensional manifolds, uniting these topics under the study of discrete subgroups of SL(2,C), inspired by Margulis's foundational work.

Contribution

It reviews recent progress in the theory of Kleinian groups and their relation to 3-manifolds, expanding on Margulis's foundational ideas in the context of SL(2,C).

Findings

Recent developments in Kleinian group theory

Connections between Kleinian groups and 3-manifolds

Advances influenced by Margulis's work

Abstract

Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title of Theory of discrete subgroups of $\operatorname{SL}_2(\mathbb C)$." In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works.