3rd bounded cohomology and Kleinian groups (3)

Theorem A. (Cohomological nearness implies geometric sameness). For a given $i_0>0$, let $(M_0,f_0)$ be any doubly-degenerate element of $AH_+(\Sigma_g)$ with $\mathrm{inj}(M_0)\geq i_0$. Then there exists a constant $\epsilon(g,i_0)>0$ depending only on $g$ and $i_0$ so that any element $(M_1,f_1)$ of $AH_+(\Sigma_g)$ satisfying $\parallel [f_0^*(\omega_{M_0})] - [f^*_1(\omega_{M_1})]\parallel<\epsilon$ in $H^3_b(\Sigma;\Bbb R)$ is equal to $(M_0,f_0)$ in $AH_+(\Sigma_g)$.

Corollary B (Geometric nearness does not imply cohomological nearness). Suppose that $\{M_n,f_n\}_{n=1}^\infty$ is a sequence in $H_+(\Sigma_g)$ such that each $(M_n,f_n)$ is not doubly-degenerate. then, $\{[f_n^*(\omega_{M_n})]_B\}_{n=1}^\infty$ contains no subsequences converging in $HB^3(\Sigma_g;\Bbb R)$ to the (induced) fundamental class of doubly-degenerate elements of $AH_+(\Sigma_g)$.

Theorem C. Suppose $\Gamma$ is a topologically tame Kleinian group such that the volume of $M_\Gamma$ is infinite. Then $[\omega_\Gamma] = 0$ in $H^3_b(M_\Gamma;\Bbb R)$ if and only if $\Gamma$ is either elementary or geometrically finite. If $[\omega_\Gamma]\neq 0$ in $H^3_b(M_\Gamma;\Bbb R)$, then $\parallel \omega_\Gamma\parallel = v_3$, so in particular, $[\omega_\Gamma]_B\neq 0$ in $HB^3(M_\Gamma;\Bbb R)$.