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Relations in bounded cohomology
Journal of Topology  (IF1.582),  Pub Date : 2020-05-04, DOI: 10.1112/topo.12150
James Farre

We explain some interesting relations in the degree 3 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 dependencies between the ‘geometric’ bounded classes defined by the volume cocycle with bounded geometry. We obtain a mapping class group invariant Banach subspace of the reduced degree 3 bounded cohomology with explicit topological generating set and describe all linear relations.