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

2025/07/14 by Tim Berland, Berland, Tim
Mathematics · #58J52 #Advanced Differential Equations and Dynamical Systems #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2507.10339

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 λ-strongly acyclic representations, which we define and show exists in plenitude for any λ>0. The motivation is possible applications to torsion in the cohomology of arithmetic groups.

Citations

• Analytic torsion for arithmetic locally symmetric manifolds and approximation of L²-torsion
• Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps
• Analytic torsion of arithmetic quotients of the symmetric space SL(n,R)/SO(n)
• Survey on approximating L²-invariants by their classical counterparts: Betti numbers, torsion invariants and homological growth
• Bounds for global coefficients in the fine geometric expansion of\n Arthur's trace formula for GL(n)
• On torsion in the cohomology of locally symmetric varieties
• On the growth of Betti numbers in p-adic analytic towers
• Analytic torsion and L²-torsion of compact locally symmetric manifolds
• The asymptotic growth of torsion homology for arithmetic groups
• L²-invariants of locally symmetric spaces

Related

• Approximation of L²-analytic torsion for arithmetic quotients of the symmetric space SL(n,ℝ)/SO(n)
• Analytic torsion of arithmetic quotients of the symmetric space SL(n,R)/SO(n)
• Analytic torsion for arithmetic locally symmetric manifolds and approximation of L²-torsion
• On the growth of torsion in the cohomology of arithmetic groups
• On the growth of torsion in the cohomology of arithmetic groups