back to N-Cube Days XXII

16:30 - 17:00 Tim With Berland (University of Copenhagen)

Title: Asymptotics of analytic torsion for congruence quotients of SL(n,R)/SO(n)

Abstract: Analytic torsion is an invariant of Riemannian manifolds introduced by Ray and Singer in the 70's, defined as an analytic analogue of the topological invariant Reidemeister torsion.
More recently, Bergeron and Venkatesh used analytic torsion and its analogy to Reidemeister 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. One of the main ingredients is an approximation theorem, expressing L^2-torsion as a limit using analytic torsion.

To extend these results for non-cocompact arithmetic groups, Matz and Müller in recent work defined analytic torsion for congruence quotients of SL(n,R)/SO(n), and later for general arithmetic locally symmetric spaces, and proved the analogous approximation theorem.
In this talk, we provide a rate of convergence result in the non-cocompact setting, which conjecturally implies stronger bounds on torsion homology.