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APPROXIMATION OF $L^{2}$-ANALYTIC TORSION FOR ARITHMETIC QUOTIENTS OF THE SYMMETRIC SPACE $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$

Part of: Partial differential equations on manifolds; differential operators Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  14 February 2018

Jasmin Matz  and
Werner Müller
Jasmin Matz
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Givat Ram, Jerusalem 9190401, Israel ([email protected])
Werner Müller
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Allee 60, D – 53115 Bonn, Germany ([email protected])
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Abstract

In [31] we defined a regularized analytic torsion for quotients of the symmetric space $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behavior of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^{2}$-analytic torsion.

Keywords

analytic torsionlocally symmetric spacestrace formula

MSC classification

Primary: 58J52: Determinants and determinant bundles, analytic torsion
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 307 - 350
Copyright
© Cambridge University Press 2018

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