扩张高维拟上圈
Extending higher dimensional quasi-cocycles
摘要：
设G是Gromov双曲空间上允许非初等非线性作用的群（例如，非初等相对双曲群，闭双曲曲面的映射类群，或Out（F_n）（n>1））.我们证明了在3度下，实系数G的有界上同调是无穷维的。我们的证明是基于扩展到更高程度的赫尔和Osin最近的结果。也就是说，如果H是G的双曲嵌入子群，V是任意G-模，则H上的任意n-拟上圈都可以扩张到G.此外，我们表明，我们的扩展检测双曲嵌入子群的嵌入的几何形状，在适当的意义。
英文摘要：
Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any G-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups, in a suitable sense.
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