Workshop on Representation Theory and Analysis on Locally Symmetric Spaces

Abstract: Let $R>0$. The R-thin part of a locally symmetric space is defined as the set of points where the R-ball is not isometric to an R-ball in the universal cover. Recently Abert et al. showed that if X is a higher rank symmetric space whose isometry group has Kazhdan's property (T), then the volume of the R-thin part of finite volume quotients M of X is asymptotically $o(Vol(M))$. It is conjectured that the same holds true for congruence arithmetic quotients of rank-1 symmetric spaces. In this talk I will present a solution (joint work with Jean Raimbault) of this conjecture for congruence arithmetic hyperbolic orbifolds of dimension 2 and 3. I will discuss some applications to topology and the relation of this problem to the limit multiplicity property for sequences of arithmetic congruence lattices.