Number-theoretic techniques in the theory of Lie groups and differential geometry

Abstract

The aim of this article is to give a brief survey of the results obtained in the series of papers [17]- [21]. These papers deal with a variety of problems, but have a common feature: they all rely in a very essential way on number-theoretic techniques (including p-adic techniques), and use results from algebraic and transcendental number theory. The fact that number-theoretic techniques turned out to be crucial for tackling certain problems originating in the theory of (real) Lie groups and differential geometry was very exciting. We hope that these techniques will become an integral part of the repertoire of mathematicians working in these areas.To keep the size of this article within a reasonable limit, we will focus primarily on the paper [21], and briefly mention the results of [17]- [20] and some other related results in the last section. The work in [21], which was originally motivated by questions in differential geometry dealing with length-commensurable and isospectral locally symmetric spaces (cf. §1), led us to define a new relationship between Zariskidense subgroups of a simple (or semi-simple) algebraic group which we call weak commensurability (cf. §2). The results of [21] give an almost complete characterization of weakly commensurable arithmetic groups, but there remain quite a few natural questions (some of which are mentined below) for general Zariski-dense subgroups. We hope that the notion of weak commensurability will be useful in investigation of (discrete) subgroups of Lie groups, geometry and ergodic theory.