Integral Foliated Simplicial Volume of Aspherical Manifolds

TL;DR

This paper explores the relationships between various forms of simplicial volume in aspherical manifolds, establishing conditions under which they coincide or differ, and connecting these concepts to ergodic theory and homology growth.

Contribution

It introduces the integral foliated simplicial volume, analyzes its properties, and demonstrates its equivalence with classical simplicial volume in specific cases using ergodic theory.

Findings

Integral foliated simplicial volume is monotone with respect to weak containment of actions.

For hyperbolic 3-manifolds and certain aspherical manifolds, all volume variants coincide.

In higher dimensions, integral foliated simplicial volume and classical simplicial volume differ.

Abstract

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth. Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case). However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4.