Length functions on mapping class groups and simplicial volumes of mapping tori

TL;DR

This paper introduces filling volume invariants on the mapping class group of a manifold, linking them to simplicial volumes of mapping tori, and explores their properties and implications for hyperbolic volume and norm equivalences.

Contribution

It defines filling volumes as new invariants on the mapping class group, relating them to simplicial volumes and analyzing their properties and applications.

Findings

Real filling volume equals the simplicial volume of the mapping torus.

Integral filling volume is at least the stable integral simplicial volume.

Hyperbolic volume of mapping tori is not subadditive with respect to monodromy.

Abstract

Let $M$ be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group $\mathrm{MCG}(M)$ of $M$, which are defined in terms of filling norms on the space of singular boundaries on $M$, both with real and with integral coefficients. We show that filling volumes are length functions on $\mathrm{MCG}(M)$, we prove that the real filling volume of a mapping class $f$ is equal to the simplicial volume of the corresponding mapping torus $E_f$, while the integral filling volume of $f$ is not smaller than the stable integral simplicial volume of $E_f$. We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of $3$-dimensional mapping tori is not subadditive with respect to their monodromy, and that the real and the integral filling norms on integral boundaries are often non-biLipschitz equivalent.