Relations in Bounded Cohomology

TL;DR

This paper explores the relations in degree three bounded cohomology of surface groups, revealing how geometric and topological properties influence cohomological classes and their linear dependencies.

Contribution

It introduces new relations among bounded fundamental classes for surface groups, especially in degenerate and geometrically finite cases, with explicit descriptions of their linear dependencies.

Findings

Quasi-isometric faithful Kleinian surface group representations have identical bounded fundamental classes.

Differences of singly degenerate classes with bounded geometry relate to doubly degenerate classes.

The relations fully describe linear dependencies among geometric bounded classes via the volume cocycle.

Abstract

We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is degenerate, while the other end is geometrically finite. We also show that a difference of two singly degenerate classes with bounded geometry is boundedly cohomologous to a doubly degenerate class, which has a nice geometric interpretation. Finally, we explain that the above relations completely describe the linear dependences between the `geometric' bounded classes defined by the volume cocycle with bounded geometry. We obtain a mapping class group invariant Banach sub-space of the reduced degree three bounded cohomology with explicit topological generating set and describe all linear relations.