Bounded cohomology and optimal separating constant for representations

Abstract

Let $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma ⟶M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\Gamma ={\pi }_1(\Sigma )$, let ${\rho }_M:\Gamma ⟶{\mathrm{PSL}}_2(\mathbb{C})$ be the holonomy of $M$ and $\rho :\Gamma ⟶{\mathrm{PSL}}_2(\mathbb{C})$ any representation in ${\mathrm{PSL}}_2(\mathbb{C})$. We will show that, if $\rho$ is discrete and non-faithful, then \[ \|[\mathrm{Vol}(ρ)]-[\mathrm{Vol}(ρ_M)]\|_\infty\geq \boldsymbol{v}_3 \] holds, where $[\mathrm{Vol}(\rho )]$ denotes the bounded fundamental class of $\rho$ in the bounded cohomology $H_b^3(\Gamma ,\mathbb{R})$ of $\Gamma$ and ${\mathbf{v}}_3$ is the volume of a regular ideal 3-simplex in ${\mathbb{H}}^3$. As an application, we present a rigidity theorem for ${\rho }_M$ in the set of representations $\rho$ of $\Gamma$ in ${\mathrm{PSL}}_2(\mathbb{C})$ in terms of $[\mathrm{Vol}(\rho )]$. The rigidity theorem implies that ${\mathbf{v}}_3$ is the optimal separating constant.

11 pages