Asymptotics of analytic torsion for hyperbolic three-manifolds
type: Item
instance of: scholarly article
title: Asymptotics of analytic torsion for hyperbolic three-manifolds
zbMATH Open document ID: 1433.58028
DOI: 10.4171/CMH/466
author: Jean Raimbault
published in: Commentarii Mathematici Helvetici
publication date: 2019-11-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.3161
review text: In this heavy technical article, the author proves limit theorems for a sequence of regularized analytic torsions on certain non-compact but finite-volume hyperbolic $3$-manifolds. More precisely, let $M_n$ be a sequence of certain finite-volume hyperbolic $3$-manifolds (i.e. arise by dividing the hyperbolic $3$-space with appropriate torsion-free lattices), being all diffeomorphic to $T^2 \times [1, +\infty )$ where $T^2$ is the flat $2$-torus, satisfying various kind of technical conditions listed up in Section 1.3 in the article. Let $V=V(n_1, n_2)$ with $n_1\not= n_2$ be the irreducible representation on $S^{n_1}{\mathbb C}^2\otimes S^{n_2}\overline{\mathbb{C}}^2$ of $\mathrm{SL}(2,\mathbb{C})$. Consider the $V$-valued \textit{regularized analytic torsion} $T_R(M_n;V)$ of $M_n$ regarded as a quotient of the hyperbolic $3$-space; likewise, let $T^{(2)}(M_n;V)$ denote the $V$-valued \textit{$L^2$-analytic torsion} of $M_n$ and put $t^{(2)}(V):=\frac{\log T^{(2)}(M_n;V)}{\mathrm{vol} M_n}$ for its universal (i.e. independent of $n$) term identified with the $L^2$-analytic torsion of the hyperbolic $3$-space itself (for their precise definitions based on the Selberg trace formula cf. Section 5.3 in the article). The author's main result is that \[ \lim_{n\rightarrow +\infty}\frac{\log T_R(M_n;V)}{\mathrm{vol} M_n} =t^{(2)}(V) \] (see Theorem A or Theorem 5.1 in the article) together with a variant (see Theorem B or Theorem 6.2) and a corollary (see Proposition C in the article) of this statement.
reviewed by: Gábor Etesi
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 7132396
zbMATH Keywords: analytic torsion
zbMATH Keywords: hyperbolic manifolds with cusps
zbMATH Keywords: Selberg trace formula
MaRDI profile type: Publication
arXiv ID: 1212.3161
cites work: On the growth of \(L^2\)-invariants for sequences of lattices in Lie groups
cites work: Analytic torsion and R-torsion of Witt representations on manifolds with cusps
cites work: Lectures on hyperbolic geometry
cites work: The asymptotic growth of torsion homology for arithmetic groups
cites work: Q4002484
cites work: ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS
cites work: Q4023901
cites work: Q4705361
cites work: Injectivity radii of hyperbolic integer homology 3-spheres
cites work: An anomaly formula for Ray-Singer metrics on manifolds with boundary
cites work: On the gluing formula for the analytic torsion
cites work: A torsion Jacquet-Langlands correspondence
cites work: Analytic torsion and the heat equation
cites work: Q4368847
cites work: Geometry of growth: approximation theorems for \(L^2\) invariants
cites work: Q4888825
cites work: Automorphic forms on semisimple Lie groups. Notes by J. G. M. Mars.
cites work: Q4790111
cites work: Analytic and topological torsion for manifolds with boundary and symmetry
cites work: Approximating \(L^ 2\)-invariants by their finite-dimensional analogues
cites work: Q3148984
cites work: \(L^2\)-torsion of hyperbolic manifolds of finite volume
cites work: On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds
cites work: Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds
cites work: Analytic Torsion and R-Torsion for Unimodular Representations
cites work: The Asymptotics of the Ray-Singer Analytic Torsion of Hyperbolic 3-manifolds
cites work: Analytic torsion of complete hyperbolic manifolds of finite volume
cites work: The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume
cites work: Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps
cites work: Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps
cites work: Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups
cites work: A GLUING FORMULA FOR THE ANALYTIC TORSION ON HYPERBOLIC MANIFOLDS WITH CUSPS
cites work: R-torsion and the Laplacian on Riemannian manifolds
cites work: On torsion in the cohomology of locally symmetric varieties
cites work: Q3918186
OpenAlex ID: W2976200345
description: scientific article