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.
Teruaki Kitano. Takayuki Morifuji. Mitsuhiko Takasawa. "$L^2$-torsion invariants and homology growth of a torus bundle over $S^1$." Proc. Japan Acad. Ser. A Math. Sci. 79 (4) 76 - 79, April 2003. https://doi.org/10.3792/pjaa.79.76