3rd bounded cohomology and Kleinian groups - Soma

게시글 주소: https://i.orbi.kr/00068693895

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 $HB^3(M_\Gamma;\Bbb R)$.

Theorem D. Let $M,M'$ be hyperbolic 3-manifolds with markings $\iota:\Sigma\to M$, $\iota':\Sigma\to M'$ respectively. Suppose that either the $(+)$ or $(-)$-end $\mathcal{E}$ of $M$ with respect to $\iota(\Sigma)$ is totally degenerate. If

$$\parallel\iota^*([\omega_M]) - \iota'^*([\omega_{M'}])\parallel<v_3$$

holds in $H^3_b(\Sigma;\Bbb R)$, then there exists a marking and orientation-preserving homeomorphism $\varphi_0:M\to M'$ and a neighborhood $E$ of $\mathcal{E}$ such that $\varphi_0|_E:E\to E' = \varphi_0(E)$ is bilipschitz. In particular, $\varphi_0$ defines the bijection between the components of $E_{cusp}$ and those of $E'_{cusp}$.

Corollary E. Under the assumptions in theorem A, suppose moreover that all genuine ends of $M$ are simply degenerate. Then $\varphi$ is properly homotopic to an isometry. In particular, $\iota^*([\omega_M]) = \iota'^*([\omega_{M'}])$ in $H^3_b(\Sigma,\Bbb R)$.

Theorem F. Let $M$ be an oriented hyperbolic 3-manifold with a marking of $\Sigma$. Suppose that there exists an orientation-preserving homeomorphism $\varphi$ from $M$ to another hyperbolic $M'$ inducing a bijection between the components of $M_{cusp}$ and those of $M'_{cusp}$. If

$$\parallel [\omega_M] - \varphi^*([\omega_{M'}])\parallel <v_3$$

holds in $H^3_b(\Sigma;\Bbb R)$, then $\varphi$ is properly homotopic to a bilipschitz map.

Theorem G. Let $G$ be a group, and let $\Gamma_1,\ldots,\Gamma_n$ be topologically tame Kleinian groups admitting isomorphisms $\varphi_i:G\to\Gamma_i$. Suppose that, for each $M_{\Gamma_i}$, only one end $E_i$ of $M_{\Gamma_i}$ is geometrically infinite. Let $\Lambda_i$ be a subgroup (uniquely determined up to conjugate) of $\Gamma_i$ corresponding to $E_i$. If, for every integers $i,j$ with $1\leq i\leq j\leq n$, there exists $\lambda_{ij}\in\Lambda_j$ such that $\varphi_i^{-1}(\Lambda_i)\cap C_\infty(\varphi_j^{-1}(\lambda_{ij})) = \emptyset$, then $[\varphi_1^*(\omega_{\Gamma_1})]_B,\ldots,[\varphi^*_n(\omega_{\Gamma_n})]_B$ are linearly independent in $HB^3(G;\Bbb R)$. Here, $C_\infty(g) = \bigcup_{n=1}^\infty C(g^n)$ where $C(g) = \{hgh^{-1}:h\in G\}$.