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\(L^ 2\)-torsion invariants and homology growth of a torus bundle over \(S^ 1\). (English) Zbl 1057.57015

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 T. Kitano, T. Morifuji and 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.

MSC:

57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57M05 Fundamental group, presentations, free differential calculus
58J52 Determinants and determinant bundles, analytic torsion

Citations:

Zbl 1068.57021
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References:

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