Analytic torsion is an invariant of compact Riemannian manifolds introduced by Ray and Singer in the 70's, defined as an analytic analogue of the topological invariant Reidemeister torsion. In 2013, Bergeron and Venkatesh used analytic torsion to describe the growth of torsion in the homology of cocompact arithmetic groups. This has number-theoretic significance by the Ash conjecture, partially proven by Scholze, relating torsion in the homology of arithmetic groups to the existence of Galois representations.

To extend these results for non-cocompact arithmetic groups, Matz and Müller in recent work defined analytic torsion and showed its limiting behaviour for congruence quotients of SL(n,R)/SO(n), and later for general arithmetic locally symmetric spaces. Furthermore, Müller and Rochon proved a link to torsion homology for non-compact hyperbolic manifolds of finite volume, allowing them to obtain similar results as Bergeron-Venkatesh in this setting.

The results in all the above cases are obtained by showing that the analytic torsion approximates the L^2-torsion, and thus stronger results are available if this limiting behavior is better understood. In this talk, we will discuss ongoing work on providing asymptotics of the analytic torsion of congruence quotients of SL(n,R)/SO(n).