Borel and volume classes for dense representations of discrete groups

Abstract

We show that the bounded Borel class of any dense representation ρ : G − → PSLn C is non-zero in the degree three bounded cohomology and has maximal Gromov semi-norm, for any countable discrete group G. For n = 2, the Borel class is equal to the 3-dimensional hyperbolic volume class. We show that the volume classes of dense representations ρ : G − → PSL 2 C are uniformly separated in semi-norm from any other representation ρ ′ : G − → PSL 2 C for which there is a subgroup H ≤ G on which ρ is still dense but ρ ′ is discrete and faithful or indiscrete but not Zariski dense. We show that the Banach subspace of reduced bounded cohomology that is the closure of volume classes of dense representations has dimension equal to the cardinality of the continuum.