\(L^2\)-torsion invariants of a surface bundle over \(S^1\)
type: Item
instance of: scholarly article
title: \(L^2\)-torsion invariants of a surface bundle over \(S^1\)
zbMATH Open document ID: 1068.57021
author: Teruaki Kitano
author: Takayuki Morifuji
author: Mitsuhiko Takasawa
published in: Journal of the Mathematical Society of Japan
publication date: 2004-10-27T00:00:00Z
review text: Let \(\Sigma\) be a compact oriented smooth surface of positive genus with one boundary component. Let \(W\) be a compact oriented \(3\)-manifold that fibers over the circle with fiber \(\Sigma\). In analogy with the definition of the combinatorial \(L^2\)-torsion \(\tau(W)\in\mathbb R\), the authors use the quotient of \(\pi_1(W)\) by the terms of the lower central series for \(\pi_1(\Sigma)\) to define a sequence \(\tau_k(W)\in\mathbb R\). They conjecture that \(\lim_{k\to\infty}\tau_k(W)=\tau(W)\) and verify this in some cases. Moreover, they give formulas for \(\tau_1(W)\) and \(\tau_2(W)\) using the number-theoretic notion of Mahler measure. The authors are motivated in part by the problem of approximating Gromov's simplicial volume of hyperbolic \(3\)-manifolds.
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 2110894
zbMATH Keywords: \(L^2\)-torsion
zbMATH Keywords: hyperbolic volume
zbMATH Keywords: surface bundle
zbMATH Keywords: nilpotent quotient
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191418642
OpenAlex ID: W2019592322
DOI: 10.2969/JMSJ/1191418642
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description: scientific article; zbMATH DE number 2110894