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- W3154010911 abstract "Abstract We show that the bounded Borel class of any dense representation $rho : Gto{operatorname{PSL}}_n{mathbb{C}}$ 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 $rho : Gto{operatorname{PSL}}_2{mathbb{C}}$ is uniformly separated in semi-norm from any other representation $rho ^{prime}: Gto{operatorname{PSL}}_2 {mathbb{C}}$ for which there is a subgroup $Hle G$ on which $rho $ is still dense but $rho ^{prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${mathbb{H}}^3cup partial{mathbb{H}}^3$. 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." @default.
- W3154010911 created "2021-04-26" @default.
- W3154010911 creator A5054486196 @default.
- W3154010911 date "2021-04-19" @default.
- W3154010911 modified "2023-10-06" @default.
- W3154010911 title "Borel and Volume Classes for Dense Representations of Discrete Groups" @default.
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- W3154010911 doi "https://doi.org/10.1093/imrn/rnab078" @default.
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