Integral foliated simplicial volume of hyperbolic 3-manifolds

TL;DR

This paper introduces the integral foliated simplicial volume, proves a proportionality principle for aspherical manifolds, and computes this volume for hyperbolic and Seifert 3-manifolds, linking it to stable simplicial volume.

Contribution

It establishes a proportionality principle for integral foliated simplicial volume and computes it explicitly for hyperbolic and Seifert 3-manifolds, advancing understanding of geometric invariants.

Findings

Proportionality principle for integral foliated simplicial volume of aspherical manifolds.

Explicit computation of the volume for hyperbolic 3-manifolds.

Calculation of the volume for Seifert 3-manifolds.

Abstract

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical manifolds and give refined upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is complemented by the calculation of the integral foliated simplicial volume of Seifert 3-manifolds.