# The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

## TL;DR

This paper constructs combinatorial volume forms to explore the zero norm subspace of bounded cohomology in acylindrically hyperbolic groups, revealing its infinite dimensionality and connecting to hyperbolic 3-manifolds and mapping tori volume bounds.

## Contribution

It introduces a new seminorm on exact bounded cohomology and demonstrates the infinite dimensionality of the zero norm subspace for acylindrically hyperbolic groups.

## Key findings

- The zero norm subspace in degree 3 is infinite dimensional.
- Combinatorial volume forms define non-trivial classes in bounded cohomology.
- A cohomological proof of a volume bound for mapping tori is provided.

## Abstract

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichm\"uller translation distance.