Bounded cohomology and negatively curved manifolds

TL;DR

This paper investigates the properties of the bounded fundamental class in the top-dimensional bounded cohomology of negatively curved manifolds, revealing conditions under which it vanishes, especially in geometrically finite and rank-one locally symmetric spaces.

Contribution

It establishes new criteria for the vanishing of the bounded fundamental class in negatively curved manifolds, linking it to geometric finiteness and bounded differential forms.

Findings

Bounded fundamental class vanishes for geometrically finite manifolds.

In rank-one locally symmetric spaces, vanishing is equivalent to the volume form being a bounded differential.

Provides a characterization connecting bounded cohomology and geometric properties.

Abstract

We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of $M$ vanishes if $M$ is geometrically finite. Furthermore, when $M$ is a $\mathbb{R}$-rank one locally symmetric space, we show that the bounded fundamental class of $M$ vanishes if and only if the Riemannian volume form on $M$ is the differential of a bounded differential form on $M$.