$L^2$-torsion invariants and homology growth of a torus bundle over $S^1$

Abstract

We introduced an infinite sequence of $L^2$-torsion invariants for surface bundles over the circle in [4]. 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.