Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume

TL;DR

This paper explores the Lipschitz simplicial volume for non-positively curved manifolds, establishing key inequalities and principles that extend Gromov's volume comparison theorem and reveal vanishing results for certain locally symmetric spaces.

Contribution

It introduces a metric version of simplicial volume, proves a proportionality principle and product inequality, and extends Gromov's volume comparison theorem to new classes of manifolds.

Findings

Proportionality principle for Lipschitz simplicial volume

Product inequality for Lipschitz simplicial volume

Vanishing of ordinary simplicial volume for certain non-compact locally symmetric spaces

Abstract

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of non-compact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of non-compact locally symmetric spaces with finite volume of Q-rank at least 3 is zero.