2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169100The 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.This is a slightly revised version of a survey article published in the Proceedings of the International Congress of Chinese Mathematicians held in 2007Differential GeometryNumber Theorytext