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.

中文翻译：

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有界同调中的关系

我们解释了表面组的3度有界同调性中的一些有趣关系。具体来说，我们表明，如果两个忠实的Kleinian曲面组表示是准等距的，则它们在有限同调性中的有限基类是相同的。在一端退化的同时另一端在几何上有限的情况下，这是新颖的。我们还表明，具有有限几何形状的两个单退化类的差异与双退化类具有有限的同调性，而双退化类具有很好的几何解释。最后，我们解释上述关系完全描述了由体积cocycle与有界几何定义的“几何”有界类之间的线性相关性。