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

TL;DR

This paper surveys recent results on length commensurability and isospectrality of locally symmetric spaces, introducing the concept of weak commensurability and employing p-adic and number theoretic methods.

Contribution

It introduces the notion of weak commensurability for Zariski-dense subgroups and demonstrates its strong implications for arithmetic subgroups using advanced number theory techniques.

Findings

Weak commensurability has strong consequences for arithmetic subgroups.

p-adic techniques are effective in studying geometric properties of Lie groups.

The survey connects geometric, algebraic, and number theoretic methods.

Abstract

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak commensurability has surprisingly strong consequences. Our proofs make use of p-adic techniques and results from algebraic and transcendental number theory.