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오리톢 [902596] · MS 2019 (수정됨) · 쪽지

2024-07-07 00:58:09
조회수 119

Klein-Maskit combination theorems

게시글 주소: https://image.orbi.kr/00068655489

Klein combination theorem. Suppose $G_1,G_2$ are two Kleinian groups with fundamental domains $D_1,D_2\subset\hat{\Bbb C}$ such that $\mathrm{int} D_1\supset\hat{\Bbb C} - D_2$ and $\mathrm{int} D_2\supset\hat{\Bbb C} - D_1$. In particular, $D_1$ and $D_2$ overlaps and the limit sets of $G_1$ and $G_2$ are disjoint. Then the subgroup $G = \langle G_1,G_2\rangle$ generated by $G_1$ and $G_2$ is a Kleinian group that is isomorphic to the free product $G_1\ast G_2$. The domain $D = D_1\cap D_2$ is a fundamental domain for the action of $G$ on $\hat{\Bbb C}$.




Maskit combination theorem은 특정 정리를 말하는 것이 아니라, 어떤 성질을 만족하는 두개의 Kleinian group으로 generated 되는 group이 어떻게 생겼는지 알 수 있는 상황을 말한다. 이 경우에는 우리는 두개의 Kleinian group을 combine했다고 표현한다. 이렇기에 Maskit combination theorem은 어마어마하게 많은데, 그 중에서 제일 중요하기도 하고 내가 이해한/와닿는 combination theorem 2가지만 소개한다. 만약 내 이해가 깊어진다면, 혹은 필요성을 느끼게 된다면 그때 추가하기로 한다.


The first Maskit combination theorem. Let $G_1$ and $G_2$ be a pair of Kleinian groups such that $H<G_1\cap G_2$. Suppose that $H$ is quasi-Fuchsian group such that $\hat{\Bbb C} - \Lambda(H) = \Omega_1\cup\Omega_2$. Assume that the domain $\Omega_j$ is precisely invariant under $H$ in $G_j$ for $j = 1,2$.

- Then the group $G$ generated by $G_1$ and $G_2$ is a Kleinian group and isomorphic to $G_1\ast_H\ast G_2$.

- If $G_1$ and $G_2$ are geometrically finite then so is $G$.

- The surface $S(G) = \Omega(G)/G$ is naturally conformally equivalent to $(S(G_1) - \Omega_1/H)\cup (S(G_2) - \Omega_2/H)$.

- Under the isomorphism $G\to G_1\ast_H\ast G_2$, the image of any parabolic element of $G$ is either conjugate to one of the groups $G_1,G_2$ or commutes with a parabolic element of a conjugate of $H$.


The second Maskit combination theorem. Let $G_0$ be a Kleinian group such that $H_1,H_2\subset G_0$, where $H_j$ are quasi-Fuchsian that stabilize different connected components $\Omega_1,\Omega_2$ of $\Omega(G_0)$. Let $\gamma\in\mathrm{PSL}_2\Bbb C$ be an element such that $\gamma(\Omega_1) = \hat{\Bbb C} - \mathrm{cl}(\Omega_2)$ and $\gamma H_1\gamma^{-1}=  H_2$ induces an isomorphism $\phi$ of $H_1$ and $H_2$.

- Then the group $G$ generated by $G_0$ and $\gamma$ is isomorphic to the HNN-extension $G_0\ast_{\phi}$ of $G_0$ via $\phi$.

- If $G_0$ is geometrically finite then so is $G$.

- The surface $S(G) = \Omega(G)/G$ is naturally conformally equivalent to $S(G_0) - (\Omega_1/H_1\cup\Omega_2/H_2)$.

- Under the isomorphism $G\to G_0\ast_{\phi}$, the image of any parabolic element in $G$ is either conjugate to the group $G_0$ or it commutes with a parabolic element of a conjugate of $H_1$.




















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