  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.
