\(L^ 2\)-torsion invariants and homology growth of a torus bundle over \(S^ 1\). (Q1432827)

The authors recently introduced an infinite sequence \((\tau_{k})_{k\in\mathbf N}\) of \(L^{2}\)--torsion invariants for surface bundles over the circle [see \textit{T. Kitano, T. Morifuji} and \textit{M. Takasawa}, J. Math. Soc. Japan 56, No. 2, 503--518 (2004; Zbl 1068.57021)]. This note concerns punctured torus bundles over the circle. The authors prove that the first invariant \(\tau_{1}\) is determined by the asymptotic behavior of the order of the first homology group of the cyclic coverings and that the second invariant \(\tau_{2}\) is always zero.