Residual finiteness and discrete subgroups of Lie groups

TL;DR

This paper explores the residual finiteness of discrete subgroups in real Lie groups, reviewing known results and proposing questions to advance understanding, especially for lattices.

Contribution

It provides a comprehensive overview of existing results on residual finiteness and introduces new questions to deepen the understanding of discrete subgroups in Lie groups.

Findings

Summarizes known positive and negative results on residual finiteness.

Identifies open questions that could lead to a complete classification for lattices.

Highlights the importance of residual finiteness in the structure of discrete subgroups.

Abstract

Let $G$ be a real Lie group and $\Gamma < G$ be a discrete subgroup of $G$. Is $\Gamma$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete answer for lattices.