# Analytic torsion for congruence quotients of $SL(n,R)/SO(n)$

 Workshop on Representation Theory and Analysis on Locally Symmetric Spaces Topic: Analytic torsion for congruence quotients of $SL(n,R)/SO(n)$ Speaker: Jasmin Matz Affiliation: Hebrew University Date: Wednesday, March 7 Time/Room: 11:30am - 12:30pm/Simonyi Hall 101

Abstract: The analytic torsion of a compact Riemannian manifold is a certain invariant which is defined via the Laplace-Beltrami operator. By the Cheeger-Mueller theorem it equals the Reidemeister torsion of that manifold and can therefore often be used to study properties of certain arithmetic lattices as, for example, in the recent work of Bergeron and Venkatesh. It is therefore of interest to extend the notion of analytic torsion to non-compact locally symmetric spaces. For finite volume hyperbolic manifolds this was achieved by various people over the last years, but no case of rank greater than 1 was considered. In my talk I want to explain how to define the analytic torsion for congruence quotients of the symmetric space $X=SL(n,R)/SO(n)$ with coefficients in strongly acyclic bundles. Further, I want discuss the behavior of the analytic torsion in the limit N $\rightarrow \infty$ for the spaces $G(N)\X$ with $G(N)$ the principal congruence subgroup of level $N$ in $SL(n, R)$. (Joint work with W. Mueller.)