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

In this paper we prove a sharpened asymptotic for the growth of analytic torsion of congruence quotients of SL (n, R) /SO (n) in terms of the volume. The result is based on a new bound on the trace of the heat kernel in this setting, allowing control of the large time behaviour of certain orbital integrals, as well as a careful analysis of error terms. The result requires the existence of λ-strongly acyclic representations, which we define and show exists in plenitude for any λ. The motivation is possible applications to torsion in the cohomology of arithmetic groups, although the connection in this setting is as of yet conjectural.