Abstract. We discuss the notion of weak commensurability of Zariski-dense subgroups of semi-simple algebraic groups over fields of characteristic zero, which enables one to match in a convenient was the eigenvalues of semi-simple elements of these subgroups. The analysis of weakly commensurable arithmetic groups has led to a resolution of some long-standing problems about isospectral locally symmetric spaces. This work has also initiated a number of questions in the theory of algebraic groups dealing with the characterization of absolutely almost simple simply connected algebraic groups having the same isomorphism classes of maximal tori over the field of definition. The recent results in this direction provide evidence to support a new conjectural form of rigidity for arbitrary Zariski-dense subgroups in absolutely almost simple algebraic groups over fields of characteristic zero based on the eigenvalue information (``eigenvalue rigidity").
Towards the eigenvalue rigidity of Zariski-dense subgroups
ICM-2014, Seoul (South Korea)