Abstract

Symmetric spaces play a central role in geometry, analysis, and arithmetic, due to their inherent rich and beautiful structure. A key question is the fascinating interplay between spectral properties of their Laplace operators on the one hand, and geometric, topological, and arithmetic properties on the other. Analytic torsion, and L2-analytic torsion, are refined spectral invariants which contain (partly conjecturally) a lot of deep, also arithmetic information. Other approaches use ergodic theory and what is called measurable group theory.