Relations in Bounded Cohomology

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.

arXiv comment34 pages, 5 figures, final version incorporating referee's suggestions, accepted for publication in Journal of Topology

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Cited by in corpus (3)

• A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices
• Borel and volume classes for dense representations of discrete groups
• Gambaudo--Ghys construction on bounded cohomology