Proc. Japan Acad. Ser. A Math. Sci. 79 (4), 76-79, (April 2003) DOI: 10.3792/pjaa.79.76
Teruaki Kitano, Takayuki Morifuji, Mitsuhiko Takasawa
KEYWORDS: $L^2$-torsion, hyperbolic volume, surface bundle, nilpotent quotient, 57Q10, 57M05, 46L10
We introduced an infinite sequence of $L^2$-torsion invariants for surface bundles over the circle in . In this note, we investigate in detail the first two terms for a torus bundle case. In particular, we show that the first invariant can be described by the asymptotic behavior of the order of the first homology group of a cyclic covering.