Volume entropy semi-norm

TL;DR

This paper introduces the volume entropy semi-norm in real homology, proves its equivalence to the simplicial volume semi-norm, and provides bounds on the systolic volume of homology class multiples.

Contribution

It defines a new semi-norm in homology, establishes its functorial properties, and proves its equivalence to the known simplicial volume semi-norm, answering a question by Gromov.

Findings

Volume entropy semi-norm is equivalent to simplicial volume semi-norm in all dimensions.

Established an upper bound on the systolic volume of multiples of homology classes.

Demonstrated functorial properties of the volume entropy semi-norm.

Abstract

We introduce the volume entropy semi-norm in real homology and show that it satisfies functorial properties similar to the ones of the simplicial volume. Answering a question of M. Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. We also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class.