\(L^2\)-torsion invariants of a surface bundle over \(S^1\)
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.
scientific article; zbMATH DE number 2110894