  We show that the bounded Borel class of any dense representation $\rho: G\to
\PSL_n\bC$ 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 $3$-dimensional hyperbolic volume class. Using tools from the theory of
Kleinian groups, we show that the volume class of a dense representation $\rho:
G\to \PSL_2\bC$ is uniformly separated in semi-norm from any other
representation $\rho': G\to \G$ for which there is a subgroup $H\le G$ on which
$\rho$ is still dense but $\rho'$ is discrete or indiscrete but stabilizes a
point, line, or plane in $\bH^3\cup \partial \bH^3$. We exhibit a family of
dense representations of a non-abelian free group on two letters and a family
of discontinuous dense representations of $\PSL_2\bR$, 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.
