Geometric convergence
게시글 주소: https://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.
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.
Remark. 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.
0 XDK (+0)
유익한 글을 읽었다면 작성자에게 XDK를 선물하세요.
-
비문학vs문학
-
저도 질문해주세요
-
제목에 "칼럼)"을 넣는다 그렇다고 꼭 공부 칼럼일 필요는 없다 가르치기만 하면...
-
선넘질 다 받음 24
개인정보나 특정될만한 질문 빼고 다 받음
-
7덮 0
국어(언매) 64 수학(확통) 76 영어 80 경제 39 정법 45 작수 44342...
-
9평 목표 적고가세요 46
건구스가 행운을 빌어준다네요~ 다른 메타로 돌려보자
-
결국 남김 4
아까워..
-
주예지쌤 넘옙흐네..
-
오르비가 이상해 7
아직 해도 안졌는데 이런 메타라고
-
백수 단점 0
돈만 많으면 없음
-
자꾸 대표가 환자 잡으라고 영업시킴. 원래는 지 업장이니 지가 잡아야하는데 페이보고...
-
앞으로 가성비 음식은 햄버거임
-
오랜만에 점심 먹었더니 위가 줄었나
-
이감 4 3
오늘 현장응시 했는데 다들 어떠셨는지..
-
우는 페페 달래주면서 무슨 생각을 했을까
-
오르비 너무 야해요 16
저 같은 순수 고닉은 감당할 수 없어요 ㅠㅠ
-
수능에서 태클 당한적 없나요? 무지성 도함수 연속 쓰면 나락가는 문제는 나온적 없나..
-
동생 손꾸락 잘릴뻔한썰 13
때는 필자가 초딩때였고 사촌들 놀러와서 집난장판 만들어 놓을때였음 잼민이 국룰인...
-
이모가 나한테 "야 니는 진짜 훌륭하다." 라고 말했음. 보통 술도, 담배도...
-
선넘질 ㄱㄱ
-
안녕하세요 17
-
Here, we state the equivalent formulations of...
-
ㅅㅂ
-
시험기간에 타과목 할시간이 너무 부족한데 방학때 수학 열심히하고 2학기부턴...
-
Mmmm?????
-
아니면 원래 못했던건가 몸도 아프고 말이 안나오네 ㅜㅜ
-
반박시 니말이 맞음. 근데 내말이 맞음
-
타 컨탠츠 대비 넘사벽 수준으로 재밌는데.. 나만 그럼?
-
S1 이고 수1은 정답륭 80퍼정도 수2는 65~70퍼 정돈데 수1은 김기현 커넥션...
-
에잉 몰라 5시간만 채우고 소맥으로 달려야겠다
-
나 개씹존잘임
-
7월더프수학21번질문 13
21번 제 풀이인데 왜 이렇게 풀면 안되는건거요?? 살려주ㅔ요ㅠㅜㅜㅜ
-
250628말고
-
미적분 시작시기 1
김기현 커리 타고있는 고2 정시러입니다 수2 킥오프복습+2,3기출하고있는데 수2...
-
혀녀기가 지금 뭐하는 거냐.... 이러다 내년에도 오르비에 있겠네... 수능 끝나고...
-
형누님들이 눈물 흘리면서 나처럼 살지 말라고 말할때 나도 마음이 편하지가...
-
더프국어를 손쉽게 때려잡는
-
식사해야지 점심에
-
동네친구들 다 흡연자라 하아 아직까진 참는중
-
사탐런 추천 7
고2인데 암기량 많은과목이 진짜 안맞아요 개념 상대적으로 적고 개념파악후 분석해서...
-
수학 3년째 3,2왔다갔다 한다. 2등급 안정 되려면 뭘 해야될까. 개때잡 거의다...
-
혹시 n수생 분들중에 국어 6모(혹은 9모)만 갑자기 망하셨던 경험 있으신분...
-
예아
-
님들 그거 앎? 8
9모 60일 남음
-
제목 그래도 성적이 너무 안올라서요... 2월 말부터 독재 잇올에서 했고 결석이나...
-
이번6평 언매 미적 영어 생윤 사문 백분위 98 95 2 98 98 입니다.. 영어...
-
1번... ? 서로바꾸면 받았습니다지 sibal 1번이노 ㅋㅋ ?
-
왜 대체 6모는 꼴아박은거지
-
ㄱㅁ하고싶다 21
-
도서관에서 수행 책 찾는 중.. 여러분은 어떻게 지내시고 계신가요?? 내신 결과 안...
첫번째 댓글의 주인공이 되어보세요.