Asymptotic properties of MMM-classes

TL;DR

This paper investigates the geometric and topological properties of MMM-classes and surface bundles, demonstrating trivial simplicial volume under certain conditions and confirming hyperbolicity of MMM-classes, advancing understanding of surface bundle invariants.

Contribution

It proves that MMM-classes are hyperbolic and that surface bundles over amenable groups have trivial simplicial volume, providing new insights into their geometric properties.

Findings

Surface bundles over amenable groups have zero simplicial volume.

All MMM-classes are hyperbolic in Gromov's sense.

Restrictions on characteristic classes of surface bundles over products.

Abstract

We study geometric properties of characteristic classes of surfaces bundles. In particular, we show that oriented surface bundles over bases with amenable fundamental groups and dimension at least 2 have trivial simplicial volume. We show furthermore that all MMM-classes are hyperbolic in the sense of Gromov, verifying a weakened version of a conjecture due to Morita. Finally we consider surface bundles over products and restrictions on their characteristic classes