Geometric convergence
게시글 주소: https://i.orbi.kr/00068642663
Here, we state the equivalent formulations of the Geometric convergence
Group theoretic formulation (Hausdorff/Chabauty topology)
1. The geometric topology on Kleinian groups we mean giving the discrete subgroup of $\mathrm{PSL}_2\Bbb C$ the Hausdorff topology as closed subsets.
- The sequence of closed subsets $\{Y_i\}$ tends to a closed subset $Z$ in Hausdorff topology of the collection of closed subsets means (1) For every $z\in Z$, there are $y_i\in Y_i$ such that $\lim_{i\to\infty} y_i = z$. (2) For every subsequence $Y_{i_j}$, and elements $y_{i_j}\in Y_{i_j}$, if $y_{i_j}\to z$ then $z\in Z$.
In other words, $\{\Gamma_i\}\to\Gamma$ geometrically if every element $\gamma\in\Gamma$ is the limit of a sequence $\{\gamma_i\in\Gamma_i\}$ and if every accumulation point of every sequence $\{\gamma_i\in\Gamma_i\}$ lies in $\Gamma$.
Rmk. It's known that the set of closed subsets is compact with Hausdorff topology. In particular, passing to a subsequence, one may always assume that a sequence of nonelementary Kleinian groups converges geometrically.
2. Equipping a hyperbolic 3-manifold $M$ with a unit orthonormal frame $\omega$ at a base point $p$ (called a base-frame), $M$ uniquely determines a corresponding Kleinian group without up to conjugacy condition by requiring that the covering projection
$$\pi:(\Bbb H^3,\tilde{\omega})\to(\Bbb H^3,\tilde{\omega})/\Gamma = (M,\omega)$$
sends the standard frame $\tilde{\omega}$ at the origin in $\Bbb H^3$ to $\omega$.
The framed hyperbolic 3-manifolds $(M_n,\omega_n) = (\Bbb H^3,\tilde{\omega})/\Gamma_n$ converge geometrically to a geometric limit $(N,\omega) = (\Bbb H^3,\tilde{\omega})/\Gamma_G$ if $\Gamma_n$ converges to $\Gamma_G$ in the geometric topology stated in 1, i.e,
-For each $\gamma\in\Gamma_G$ there are $\gamma_n\in\Gamma_n$ with $\gamma_n\to\gamma$.
-If elements $\gamma_{n_k}$ in a subsequence $\Gamma_{n_k}$ converges to $\gamma$, then $\gamma$ lies in $\Gamma_G$.
(intrinsic) Manifold formulation
3. $(M_n,\gamma_n)$ converges to $(N,\gamma)$ geometrically if for each smoothly embedded compact submanifold $K\subset N$ containing $\omega$, there are diffeomrophism (or quasi-isometries or biLipschitz) $\phi_n:K\to (M_n,\omega_n)$ so that $\phi_n(\omega) = \omega_n$ and so that $\phi_n$ converges to an isometry on $K$ in the $C^\infty$-topology.
Rmk. Note that one can formulate the above by saying that for $\epsilon>0$, there is a sequence of isometric embeddings $\beta_i: B_{\epsilon}(\phi_i(x))\to\Bbb H^3$ from $\epsilon$-ball around $\phi_i(x)\in M_i$ so that $\beta_i\circ\phi_i$ converges to an isometric embedding of some neighborhood of $x\in N$ into $\Bbb H^3$.
4. A sequence of Kleinain groups $\Gamma_i$ converges geometrically to the Kleinain groups $\Gamma_G$ if there exists a sequence $\{r_i,k_i\}$ and a sequence of maps $\tilde{h}_i:B_{r_i}(0)\subset\Bbb H^3\to\Bbb H^3$ such that the following holds:
(1) $r_i\to\infty$ and $k_i\to 1$ as $i\to\infty$;
(2) the map $\tilde{h}_i$ is a $k_i$-bi-Lipschitz diffeomorphism onto its image, $\tilde{h}_i(0) = 0$, and for every compact set $A\subset\Bbb H^3$, $\tilde{h}_i|_A$ is defined for large $i$ and converges to the identity in the $C^\infty$-topology; and
(3) $\tilde{h}_i$ descends to a map $h_i:Z_i = B_{r_i}(p_G)\to M_i = \Bbb H^3/\Gamma_i$ is a topological submanifold of $M_G$; moreover, $h_i$ is also a $k_i$-bi-Lipschitz diffeomorphism onto its image. Here, $p_G = \pi_G(0)$ where $\pi_G:\Bbb H^3\to M_G$.
Gromov-Hausdroff formulation
5. The sequence of discrete groups $\{G_n\}$ converges polyhedrally to the group $H$ if $H$ is a discrete and for some point $p\in\Bbb H^3$, the sequence of Dirichlet fundamental polyhedra $\{P(G_n)\}$ centered at $p$ converge to $P(H)$ for $H$, also centered at $p$, uniformly on compact subsets of $\Bbb H^3$. More precisely, given $r>0$, set
$$B_r = \{x\in\Bbb H^3:d(p,x)<r\}.$$
Define the truncated polyhedra $P_{n,r} = P(G_n)\cap B_r$ and $P_r = P(H)\cap B_r$. A truncated polyhedron $P_r$ has the property that its faces (i.e. the intersection with $B_r$ of the faces of $P$) are arranged in pairs according to the identification being made to form a relatively compact submanifold, bounded by the projection of $P\cap\partial B_r$. We say that this polyhedral converges if: Given $r$ sufficiently large, there exists $N = N(r)>0$ such that (i) to each face pairing transformation $h$ of $P_r$, there is a corresponds a face pairing transformation $g_n$ of $P_{n,r}$ for all $n\geq N$ such that $\lim_{n\to\infty}g_n = h$, and (ii) if $g_n$ is a face pairing transformation of $P_{n,r}$ then the limit $h$ of any convergent subsequence of $\{g_n\}$ is a face, edge or vertex pairing transformation of $P_r$.
In other words, each pair of faces of $P_r$ is the limit of a pair of faces of $\{P_{n,r}\}$ and each convergence subsequence of a sequence of face pairs of $\{P_{n,r}\}$ converges to a pair of faces, edges, or vertices of $P_r$.
A seuqnece $\{G_n\}$ of Kleinian groups converges geometrically to a nonelementary Kleinian group if and only if it converges polyhedrally to a nonelementary Kleinian group.
Rmk. It's necessary that one needs to assume the limit group nonelementary. It's possible that the geometric limit of nonelementary Kleinian group is an elementary Kleinian group.
6. A sequence $X_k$ of metric spaces converges to a metric space $X$ in a sense of Gromov-Hausdorff if it converges w.r.t. the Gromov-Hausdorff distance. Here, Gromov-Hausdorff means the following:
Let $X$ and $Y$ be metric spaces. A triple $(X',Y',Z)$ consisting of a metric space $Z$ and its two subsets $X'$ and $Y'$, which are isometric respectively to $X$ and $Y$, will be called a realization of the pair $(X,Y)$. We define the Gromov-Hausdorff distance:
$$d_{GH}(X,Y) = \inf\{r\in\Bbb R:\text{ there exists a realization }(X',Y',Z)\text{ of }(X,Y)\text{ such that }d_H(X'.Y')\leq r\}$$
where $d_H$ is a Hausdorff distance.
addendum. A sequence of representations $\varphi_n\in AH(\Gamma)$ converges algebraically to $\varphi\in AH(\Gamma)$ if $\lim_{n\to\infty}\varphi_n(\gamma) = \varphi(\gamma)$ for each $\gamma\in\Gamma$. This is a natural topology once we view $AH(\Gamma) = \mathrm{Hom}(\Gamma,\mathrm{PSL}_2\Bbb C)/\mathrm{PSL}_2\Bbb C\subset \mathrm{Hom}(\Gamma,\mathrm{PSL}_2\Bbb C)//\mathrm{PSL}_2\Bbb C$ as an algebraic variety.
Here, $\mathrm{Hom}$ we implicitly assume it's weakly type preserving but not necessary (strongly) type preserving.
In the manifold term, one can describe the algebraic convergence as follows: Element in $AH(\Gamma)$ can be thought as a homotopy equivalence (called the marking) $h:N\to M$ where $N$ is some fixed hyperbolic 3-manifold with $\pi_1(N) = \Gamma$ such that two elements $(M,h)$ and $(M',h')$ are equivalent if there is an isometry $\psi:M\to M'$ such that $\psi\circ h\simeq h'$. Note that this is equivalent to the discrete faithful representation of $\Gamma$ to $\mathrm{PSL}_2\Bbb C$ by the $K(G,1)$-space property.
Under this view point, a sequence of marked manifolds $(M_i,h_i)$ converges algebraically to $(M,h)$ if there is a smooth homotopy equivalences $H_i: M\to M_i$ compatible with the marking that converges $C^\infty$ to local isometries on compact subsets of $M$.
It's noted that the algebraic convergence of $(M_i,h_i)$ to $(M,h)$ is guaranteed if there is a compact core $K$ of $M$ and a smooth homotopy equivalences $H_i:K\to M_i$ compatible with the markings and which are $L_i$-bilipschitz diffeomorphisms on $K$ with $L_i\to 1$.
Remark/Properties. 1. If $\rho_i:\Gamma\to\mathrm{PSL}_2\Bbb C$ is a sequence of discrete faithful representation that converges algebraically to $\rho$ and geometrically to $\Gamma_G$, then $\rho(\Gamma) = \Gamma_A\subset\Gamma_G$ because by definition, $\Gamma_A$ consists of all convergence sequences $\rho_i(g)$ for fixed $g\in\Gamma$ whereas $\Gamma_G$ contains all convergence sequences of the form $\rho_i(g_i)$ for $g_i\in\Gamma$.
2. Although after passing to a subsequence, algebraically convergence sequence implies geometric convergence, geometric convergence itself does not imply algebraic convergence.
3. Suppose a sequence of discrete faithful representations $\rho_i:\Gamma\to\mathrm{PSL}_2\Bbb C$ converge algebraically to $\rho$ and geometrically to $\Gamma_G$. Then there is not $\gamma\in\Gamma_G - \rho(\Gamma)$ with $\gamma^k\in\rho(\Gamma)$ for some $k\geq 2$. In particular, if the image $\rho(\Gamma)$ of the algebraic limit has finite index in the geometric limit $\Gamma_G$, then $\rho(\Gamma) = \Gamma_G$.
$(\because)$ Suppose there is $g\in\Gamma_G - \rho(\Gamma)$ with $g^k = \rho(\eta)$ for some $\eta\in\Gamma$ for $k\geq 2$. Since $g\in\Gamma_G$, there is a sequence $\gamma_i\in\Gamma_i$ that $\rho_i(\gamma_i)\to g$. Taking power $k$ gives
$$\lim_{i\to\infty}\rho_i(\gamma_i^k) = g^k = \rho(\eta) = \lim_{i\to\infty}\rho_i(\eta).$$
It can be shown (via nontrivial argument) that $\rho_i(\gamma_i^k) = \rho_i(\gamma)$ using the fact that $\rho_i$ converges algebraically to $\rho$. Since the representation is faithful, this implies $\gamma_i^k = \gamma$ for large $i$. It can be shown also that the set of roots $\gamma = \gamma_i^k$ is finite in general. Hence, after passing to a subsequence, $\gamma_i = \gamma_j$ for all $i,j$ so that $g\in\rho(\Gamma)$ which is a contradiction. $\square$
0 XDK (+0)
유익한 글을 읽었다면 작성자에게 XDK를 선물하세요.
-
칠리베이컨 웜볼<<원툴로 먹는데 배달로 시킬땐 9500원 이었거든요 근데 학원 주변...
-
점수보고 충격적임 ..특히 수학 국어ㅠ 하
-
아닌가
-
공통 1틀인데 27번부터 싸그리 틀려서 81..... 내가 미적을 못하는건가ㅜㅜ
-
은근 있네요? 논란 있는거 싫어하는거 아니었나 그 사람의 행보와 작품은...
-
폰이 ㅈㄴ 잘되네 내 노트북 중3때 산거라 그런지 개느렸나봐...
-
더프 성적 4
화작 확통 영어 생윤 사문 67 77 3 44 36 기록용으로 올림 9모땐 달라질 예정이라.
-
한결같이 에스파스러운것만 해줘서 좋음
-
학원은 왜 4
자발적결사체가 아닌가욮
-
서바라 그런겨?
-
한줄 요약: 터지다. 1. 국어: 15,16,17,24,34 틀 88점 언매...
-
대표적인곳이 연세대, 경희대로 알고 있는데 과탐 가산점 있으면 미적사탐으로도...
-
화1 개념 강의 3
고2고 내신대비로 화학1 공부하려고 하는데 작년에 김준 필수이론, 기출 들었었는데...
-
ㅅㅅ 미적3틀 고정인듯 ㅠㅠ
-
바이럴 왜이리많지 아는애중에 작년 기백이있는데 걔한테 물어볼까..
-
4규 n티켓 빅포텐 s1 설맞이 풀었습니다 작년 커넥션 있는데 걍 올해 나온 엔제들 푸는게 낫나요
-
유베가는길들어야함?? 듣기는0-1개 틀림 밤수라시간도없는데 그냥 구문20수에어랑...
-
걱정 덜어드림 4
항상 여러분 밑에는 제가 있으니 걱정하지 말라구 찡끗
-
독서 내신대비 2
독서는 내신대비 어떻게 해야하나요? 기출 풀면 될까요?
-
뀨루룩 0
끼룩끼룩
첫번째 댓글의 주인공이 되어보세요.