Klein-Maskit combination theorems

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.

꽤나 쉬운 Klein combination과는 다르게, Maskit combination은 약간 복잡하다. 왜 이런 형태의 combination theorem이 나왔는지는 motivating example을 보면서 이해하는 것이 가장 좋다: smooth manifold $M$이 hypersurface $S$를 기준으로 $A$와 $B$라는 component로 separate 됐다고 치자. 그러면, van Kampen에 의해서, $\pi_(M)$은 $\pi_1(A)\ast_{\pi_1(S)}\pi_1(B)$와 isomorphic하다. 이러한 $A,B,S$가 $M$의 universal cover에서 (우리의 경우에는 결국 $\Bbb H^3$가 될 것이다) 어떻게 $\pi_1(A),\pi_2(B),\pi_1(S)$와 interaction을 하는지 살펴보자.

먼저 $\tilde{A},\tilde{B},\tilde{S}$를 $A,B,S$ 각각의 universal cover라고 하자. 그러면 우리는 $\tilde{A},\tilde{B}$들의 copy들에 의해서 $\tilde{M}$이 tessellate이 된다는 것을 알 수 있고, 또한 $S$가 자체가 separating 이었기 때문에, $\tilde{S}$의 copy들로 인해서 $\tilde{A}$와 $\tilde{B}$가 $\tilde{M}$에서 separate된다는 것을 알 수 있다. 이제 $Z$라는 $\tilde{S}$의 copy하나로 separate되는 $X,Y$라는 $\tilde{A},\tilde{B}$의 representative하나를 잡자. 그러면 $X,Y$ 각각을 stabilize하는 component subgroup $G_A,G_B$를 잡을 수 있고, 당연히 각각 $\pi_1(A),\pi_1(B)$와 isomorphic하다. 또한, surface $Z$는 precisely invariant under $J = \pi_1(S)$ in both $G_A$ and $G_B$ 다. 다시 말해서, 각각의 $\gamma\in J$에 대해서, $\gamma(Z) = Z$이고, $\eta\in G_A - J$ 혹은 $\eta\in G_B - J$에 대해서, $\eta(Z) \cap Z = \emptyset$ 이다. 이제 $X^+$ (resp. $Y^+$)를 $\tilde{M} - Z$의 component들 중에 $X$ (resp. $Y$)를 포함하는 component라고 하자. 그러면, domain $X^+$는 precisely invariant under $J$ in $G_B$이고 $Y^+$는 precisely invariant under $J$ in $G_A$가 된다. 다시 말해서, $X^+$와 $Y^+$는 정확히 $J$의 action에서만 접점이 있고, 그 외에서는 전혀 접점이 없다.

The first Maskit combination theorem은 정확히 저 위의 과정을 뒤집은 것이다. 일단 group $G_A,G_B$ acting on a space $W$를 가져온 뒤에, $G_A\cap G_B >J$라는 subgroup을 포함되는 상황을 설정한다. 그러면, $W$를 $X^+$과 $Y^+$같은 domain들로 $Z$를 따라서 decompose를 하게 되고, $J$는 $Z$를 stabilize하는 상황. 그리고 $X^+,Y^+$ 또한 precisely invariant property를 갖고 있어야 한다. 그렇게 되면, $G_A$와 $G_B$로 generate되는 group은 정확히 $G_A\ast_J G_B$이고, fundamental domain은 "기대한 바" 를 얻게 된다.

The second Maskit combination theorem은 $S$가 nonseparating인 경우를 다룬다. 다시 말해서, 하나의 component에서 합치는 장면을 말하고 있다. 이 경우에는 HNN-extension이 나오는데, mapping torus와 같은 상황을 생각하면 편하다 (물론 mapping torus는 combination theorem으로 만들어지는 것은 아니다. 그냥 상황이 비슷하다는 것이다).

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 $H$.

Rmk. 가정 중에서 가장 중요한 것은, $\hat{\Bbb C} - \Lambda(H) = \Omega_1\cup\Omega_2$라는 가정으로, 우리가 $H$를 따라서 붙이려는 conformal boundary의 구조와 각도등이 정확히 $G_1$과 $G_2$에 해당되는 conformal boundary와 일치해야 한다는 의미다.

The first Maskit combination theorem (version 2). Let $\Gamma_1$ and $\Gamma_2$ be two Kleinian groups, and $C$ a topological plane in $\Bbb H^3$ whose complement consists of two components $B_1$ and $B_2$. If $B_i$ is $(J,\Gamma_i)$-invariant for $i = 1,2$ where $J = \Gamma_1\cap\Gamma_2$, meaning $g(B_i)\cap B_i = \emptyset$ for any $g\in\Gamma_i - \mathrm{stab}_{\Gamma_i}(B_i)$ and $J = \mathrm{stab}_{\Gamma_i}(B_i)$, then we have the following consequences for $\Gamma = \langle\Gamma_1,\Gamma_2\rangle$:

(1) $\Gamma$ is Kleinian. In the case $J = \{1\}$, it is represented as a free product $\Gamma = \Gamma_1\ast\Gamma_2$;

(2) Letting $p_i:\Bbb H^3\to\Bbb H^3/\Gamma_i$ for $i=1,2$ be the covering projection, we obtain a hyperbolic orbifold $N$ by asting

$$p_1(\Bbb H^3) - p_1(B_1)\text{ and }p_2(\Bbb H^3) - p_2(B_2)$$

along $p_1(C) = p_2(C) = C/J$. If $N$ is complete, $N = \Bbb H^3/\Gamma$.

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 $H_1$.

Rmk. Second combination에서 가장 중요한 가정은 $\gamma(\Omega_1) = \hat{\Bbb C} - \mathrm{cl}(\Omega_2)$ 라는 것이다. 다시 말해서, 우리가 $\gamma$를 통해서 붙이려는 conformal boundary들은 conformal structure가 같고 ($\gamma H_1\gamma^{-1} = H_2$) 그리고 붙일 때 그 "각도" 혹은 "모양" 이 같아야 한다는 것이다. 이러한 비유는 예전에 어떤 대가께서 (정확히 이 분야를 하지는 않는다) combination theorem을 말할 때 썼던 비유로, 당시에는 제대로 느껴지진 않았지만, 지금 생각해보면 이것보다 정확한 비유는 없다고 생각한다.

The second Maskit combination theorem (version 2). Let $\Gamma_0$ be a Kleinian group, $f$ an element of $\mathrm{Isom}^+(\Bbb H^3)$, and $C$ a topological plane in $\Bbb H^3$ whose complement consists of $B$ and the other complement. If $B' = \Bbb H^3 - \overline{f(B)}$ is disjoint from $\gamma(B)$ for every $\gamma\in\Gamma_0$, if $B$ is $(J,\Gamma_0)$-invariant and if $B'$ is $(fJf^{-1},\Gamma_0)$-invariant where $J = \Gamma_0\cap f^{-1}\Gamma_0 f$, then we have the following consequences for $\Gamma = \langle\Gamma_0,f\rangle$:

(1) $\Gamma$ is Kleinian. In the case $J = \{1\}$, it is represented as a free product $\Gamma = \Gamma_0\ast\langle f\rangle$.

(2) Letting $p:\Bbb H^3\to\Bbb H^3/\Gamma_0$ be the covering projection, we obtain a hyperbolic orbifold $N$ by pasting the boundaries

$$p(C) =  C/J\text{ and }p(f(C)) = f(C)/fJf^{-1}$$

of $p(\Bbb H^3) - (p(B)\cup p(B'))$. If $N$ is complete, $N = \Bbb H^3/\Gamma$.

Theorem. If a hyperbolic 3-manifold $M_\Gamma$ contains a properly embedded incompressible surface $S$, either of the following assertion is satisfied:

(1) If $S$ divides $M_\Gamma$ into $M_1$ and $M_2$, then $\Gamma$ contains subgroup $\Gamma_0\simeq\pi_1(M_1)$ and $\Gamma_2\simeq\pi_1(M_2)$ from which we can reconstruct $\Gamma$ by the first Maskit combination theorem; or

(2) If $M_0 = M_\Gamma - S$ is connected, then $\Gamma$ contains a subgroup $\Gamma_0\simeq\pi_1(M_0)$ and an element $f\in\mathrm{Isom}^+(\Bbb H^3)$ from which we can reconstruct $\Gamma$ by an application of the second Maskit combination theorem.

$(\because)$ Note that since $S$ is incompressible, any component $C$ of $\pi^{-1}(S)$ under the universal covering map $\pi:\Bbb H^3\to\Bbb H^3/\Gamma$ is simply connected. Since $S$ is not a sphere (recall Kleinian manifold is irreducible), $C$ is a topological plane in $\Bbb H^3\cup\Omega(\Gamma)$. Then $\pi^{-1}(S)$ divides $\Bbb H^3\cup\Omega(\Gamma)$ into a countable number of connected components, and we denote the two of them meeting along $C$ by $T_1$ and $T_2$. In the case when there is an element $f\in\Gamma$ such that $f(T_1) = T_2$, we apply the second Maskit combination theorem, and otherwise the first Maskit combination theorem. We set $\Gamma_1 = \mathrm{stab}_\Gamma(T_1)$ and $\Gamma_2 = \mathrm{stab}_\Gamma(T_2)$ for the first Maskit and $\Gamma_0 = \mathrm{stab}_\Gamma(T_1)$ for the second Maskit. Now it's easy to see they satisfy the assumptions of the combination theorem. $\square$

Theorem. If the domain of discontinuity $\Omega(\Gamma)$ of a torsion-free Kleinian group $\Gamma$ has a component $\Delta$ which is not simply connected, then $\Gamma$ has a nontrivial free product decomposition $(I)$ $\Gamma = \Gamma_1\ast\Gamma_2$ or $(II)$ $\Gamma = \Gamma_0\ast\langle f\rangle$, and $\Omega(\Gamma)/\Gamma$ is constructed as follows in each case respectively:

$(I)$ From each of $\Omega(\Gamma_1)/\Gamma_1$ and $\Omega(\Gamma_2)/\Gamma_2$, remove a disk and paste the resulting ones along the new boundaries, which is $\Omega(\Gamma)/\Gamma$;

$(II)$ From $\Omega(\Gamma_0)/\Gamma_0$, remove two disjoint disks and sew the resulting one on itself along the new boundaries, which is $\Omega(\Gamma)/\Gamma$.

$(\because)$ The surface $S = \Delta/\mathrm{stab}_\Gamma(\Delta)$ is compressible as $\Delta$ is not simply connected, and $\pi_1(S)\neq\{1\}$. Then by the loop theorem, there is a compression disk $D$ in $M_\Gamma$ whose boundary is a nontrivial simple closed curve $\alpha$ in $S$. By the above theorem, $\Gamma$ is $\Gamma_1\ast\Gamma_2$ or $\Gamma_0\ast\langle f\rangle$, which are constructed via the combination theorems. Further, these free products are non-trivial because $\alpha$ is nontrivial in $S$.  Let $C$ be a topological plane in $\Bbb H^3\cup\Omega(\Gamma)$ that is connected component of $\pi^{-1}(D)$ where $\pi$ is the universal cover of $M_\Gamma$. the boundary $\partial C$ is simple closed curve in $\Omega(\Gamma)$ that is a lift of $\alpha$. Then the construction of the resulting hyperbolic manifolds in the Maskit combination theorem reflects that of $\Omega(\Gamma)/\Gamma$ as in the statement above. $\square$

Klein-Maskit combination theorem for infinite type surface.