Abstract. The goal of this lecture series is to present the techniques for analyzing arithmetic and general Zariski-dense subgroups of algebraic groups that were developed in a joint work with Gopal Prasad. This work was initially motivated by the long-standing problems in geometry concerning isospectral and iso-length-spectral Riemannian manifolds. We have been able to settle some of these problems for locally symmetric spaces associated with simple real algebraic groups through introducing the notion of weak commensurability for Zariski-dense subgroups of semi-simple algebraic groups and investigating when two arithmetic groups are weakly commensurable. In order to describe various ingredients of this approach, we will include a brief review of the required notions and results from the theory of algebraic groups.

The important feature of our analysis of weak commensurability is that many techniques, and also certain results, apply not only to arithmetic but in fact to arbitrary (finitely generated) Zariski-dense subgroups. In particular, there is growing evidence to support the expectation that two simple algebraic groups containing weakly commensurable Zariski-dense subgroups must be closely related. We call this phenomenon ``eigenvalue rigidity" since weak commensurability is based on matching the eigenvalues of the elements in the subgroups. It should be pointed out that this new form of rigidity is expected to hold even when the given Zariski-dense subgroups are free groups, and the classical rigidity theorems are inapplicable. We will report on the current work in this direction which is joint with Vladimir Chernousov and Igor Rapinchuk.