Piecewise straightening and Lipschitz simplicial volume

TL;DR

This paper introduces a piecewise straightening method for singular chains to extend key properties of Lipschitz simplicial volume, including the proportionality principle and product inequality, to broader classes of Riemannian manifolds.

Contribution

The authors develop a novel piecewise straightening procedure that generalizes the proportionality principle and product inequality for Lipschitz simplicial volume to non-compact manifolds.

Findings

Extended proportionality principle to complete Riemannian manifolds with bounded sectional curvature.

Proved the product inequality for Lipschitz simplicial volume in new settings.

Provided a new proof of the proportionality principle in the compact case.

Abstract

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above. We obtain also yet another proof of the proportionality principle in the compact case by a direct approximation of the smearing map.