Ideal simplicial volume of manifolds with boundary

TL;DR

This paper introduces the ideal simplicial volume for manifolds with boundary, a new invariant that generalizes Gromov's simplicial volume by allowing ideal simplices, and explores its properties and computations.

Contribution

It defines the ideal simplicial volume, compares it with classical volume, and computes it for certain hyperbolic 3-manifolds, revealing new insights and sharper bounds.

Findings

Ideal simplicial volume is bounded above by classical simplicial volume.

It coincides with classical volume for manifolds with amenable boundary.

It can be strictly smaller for hyperbolic manifolds with geodesic boundary.

Abstract

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, the main difference being that ideal simplices are now allowed to appear in representatives of the fundamental class. We show that the ideal simplicial volume is bounded above by the ordinary simplicial volume, and that it vanishes if and only if the ordinary simplicial volume does. We show that, for manifolds with amenable boundary, the ideal simplicial volume coincides with the classical one, whereas for hyperbolic manifolds with geodesic boundary it can be strictly smaller. We compute the ideal simplicial volume of an infinite family of hyperbolic $3$-manifolds with geodesic boundary, for which the exact value of the classical simplicial volume is not known, and we exhibit examples where the ideal simplicial volume provides shaper bounds on mapping degrees than the classical simplicial volume.