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arXiv · 2507.10339

Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$

Tim Berland
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Abstract

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 bounds on the trace of the heat kernel, 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 $\lambda$-strongly acyclic representations, which we define and show exists in plenitude for any $\lambda>0$. The motivation is possible applications to torsion in the cohomology of arithmetic groups.

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Tim Berland. 2025-07-14. Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$. https://arxiv.org/abs/2507.10339

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