Bounded cohomology and topologically tame Kleinian groups (Q1362083)

The aim of this paper is to present some of the relationships between the third bounded cohomology and connections with hyperbolic 3-manifolds. Let \(\Gamma\) be a torsion-free Kleinian group, \(H^3\) a hyperbolic 3-space and \(\omega_\Gamma\) a bounded 3-cocycle on the hyperbolic 3-manifold \(M_\Gamma =H^3/ \Gamma\), defined by \(\omega_\Gamma (\sigma)= \Omega_\Gamma\) (straight \((\sigma))\) for any singular simplex \(\sigma: \Delta^3 \to M_\Gamma\); \(\Omega_\Gamma\) denotes the volume form on \(M_\Gamma\). The author studies the case where the \(\Gamma\) are topologically tame Kleinian groups and he obtains several results: Theorem. Suppose that \(\Gamma\) is a topologically tame Kleinian group such that the volume of \(M_\Gamma\) is infinite. Then \([\omega_\Gamma] =0\) in \(H^3_b (M_\Gamma; \mathbb{R})\) if and only if \(\Gamma\) is either elementary or geometrically finite. If \([\omega_\Gamma] \neq 0\) in \(H^3_b(M_\Gamma; \mathbb{R})\), then \(|[\omega_\Gamma] |= V_3\) and hence, in particular, \([\omega_\Gamma]_B \neq 0\) in \(HB^3(M_\Gamma; \mathbb{R})\). Here, \(H^*_b (X;\mathbb{R})\) denotes the bounded cohomology of the topological space \(X\) and \(HB^*(X;\mathbb{R}) =H^*_b (X;\mathbb{R})/\{[c] \in H^*_b(X;\mathbb{R}); |[c] |=0\}\).