Borel and Volume Classes for Dense Representations of Discrete Groups

Abstract

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.

A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices

“…In a more general setting, Francaviglia and Klaff [FK06] proved some similar rigidity results for their definition of volume of a representation Γ → PO(m, 1), this time assuming m ≥ n ≥ 3 (the rigidity of volume actually holds also at infinity, as proved by Francaviglia and the second author [FS18] for the real hyperbolic lattices: Moreover, the second author also showed that the rigidity holds for complex and quaternionic lattices [Sav18]). The interest in the study of volume of representations has recently grown, leading to the development of a rich literature [Poz15,KK16,Tho18,Fara,Farb].…”