Bounded cohomology and isometry groups of hyperbolic spaces
Bounded cohomology and isometry groups of hyperbolic spaces
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双曲空间的有界上同调群和等距群
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发表时间：
2005
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通讯作者：
U. Hamenstaedt
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作者：
U. Hamenstaedt
设\(X\)为任意双曲测地度量空间，设\(\Gamma\)为\(X\)的等距群\(\mathrm{Iso}(X)\)的一个可数子群。我们证明，如果\(\Gamma\)是非初等的且弱无圈的（这是一种弱的恰当性条件），那么二阶有界上同调群\(H_b^2(\Gamma,\mathbb{R})\)，\(H_b^2(\Gamma,\ell^p(\Gamma))\)（\(1 < p < \infty\)）是无穷维的。例如，我们的结果对有限型非例外曲面的映射类群的任何子群都成立，只要该子群不包含一个实质上分裂为直积的正规子群。
Let $X$ be an arbitrary hyperbolic geodesic metric space and let $Gamma$ be a countable subgroup of the isometry group ${
m Iso}(X)$ of $X$. We show that if $Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(Gamma,mathbb{R})$, $H_b^2(Gamma,ell^p(Gamma))$ $(1< p <infty)$ are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct product.