We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

中文翻译：

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离散群密集表示的 Borel 和 Volume 类

我们证明了任何稠密表示 $\rho 的有界 Borel 类： G\to{\operatorname{PSL}}_n{\mathbb{C}}$ 在三度有界上同调上是非零的，并且具有最大半范数，对于任何离散组 $G$。当$n=2$时，Borel类等于三维双曲体积类。使用克莱因群理论中的工具，我们证明了稠密表示 $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ 的体积类以半范数从任何其他表示 $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$，其中有一个子群 $H\le G$，$\rho $ 是仍然是稠密的，但 $\rho ^{\prime}$ 是离散的或不离散的，但稳定了 ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$ 中的点、线或平面. 我们展示了两个字母上的非阿贝尔自由群的密集表示族和 ${\operatorname{PSL}}_2{\mathbb{R}}$ 的不连续密集表示族，其体积类是线性独立的，并且满足一些附加性质；这些家庭的基数是连续体的。我们解释了如何使用所采用的策略来产生更高维度的非平凡体积类，这取决于具有某些拓扑和几何特性的双曲流形族的存在。