Bounded cohomology and volume rigidity of hyperbolic 3-manifolds

Abstract:

We present a rigidity theorem for hyperbolic 3-manifolds $M=\mathbb {H}^3/\Gamma$ with a Kleinian surface group $\Gamma$ in terms of the fundamental class $[\omega _M]$ in the bounded cohomology $H_b^3(M;\mathbb {R})$. Under some conditions, we show that a homeomorphism $\varphi :M\longrightarrow M’$ between complete hyperbolic 3-manifolds $M$, $M’$ is properly homotopic to a bi-Lipschitz map if the pseudo-norm of $[\omega _M]-\varphi ^*([\omega _{M’}])$ in $H_b^3(M;\mathbb {R})$ is less than the volume of a regular ideal simplex in the hyperbolic 3-space. We see that the separation constant is optimal. Finally a rigidity theorem for representations $\rho :\Gamma \longrightarrow \mathrm {PSL}_2(\mathbb {C})$ with respect to the fundamental class $[\mathrm {Vol}(\rho )]$ in $H_b^3(\Gamma ,\mathbb {R})$ is also presented.