Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds

TL;DR

This paper establishes a relationship between the vanishing of the bounded fundamental class and the positivity of the Cheeger isoperimetric constant in negatively curved manifolds with bounded geometry.

Contribution

It proves that the bounded fundamental class vanishes if and only if the Cheeger constant is positive, partially confirming a conjecture by Kim and Kim.

Findings

Bounded fundamental class vanishes iff Cheeger constant is positive for certain manifolds.

Positivity of Cheeger constant implies vanishing of the bounded volume class.

Results hold for manifolds with negative curvature bounded away from zero and infinite volume.

Abstract

We prove that for manifolds with negative curvature bounded away from $0$ of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes if and only if the Cheeger isoperimetric constant is positive. This gives a partial affirmative answer to a conjecture of Kim and Kim. Furthermore, we show that for all manifolds with negative curvature bounded away from $0$ of infinite volume, the positivity of the Cheeger constant implies the vanishing of the bounded volume class, solving one direction of the conjecture in full generality.