On hyperbolic cohomology classes

TL;DR

This paper investigates hyperbolic cohomology classes within simplicial complexes, establishing their invariance properties, linking their existence to non-amenability of fundamental groups, and exploring their relation to atoroidal classes with applications to symplectically atoroidal manifolds.

Contribution

It provides new insights into hyperbolic cohomology classes, including homological invariance, their connection to non-amenability, and clarifies their relation to atoroidal classes in degree two.

Findings

Hyperbolic classes are homologically invariant.

Existence of hyperbolic classes correlates with non-amenability.

In degree two, hyperbolic and atoroidal classes are related, impacting symplectic geometry.

Abstract

We study hyperbolic cohomology classes in the general context of simplicial complexes and prove homological invariance statements for them. We relate the existence of hyperbolic cohomology classes to the non-amenability of the fundamental group. In degree two we clarify the relation between hyperbolic and atoroidal classes, leading to an application to symplectically atoroidal manifolds.