무브

오르비

아톰

내 태그

내 태그 설정

입시
학습
생활
클럽
직업·취업
Epioptimus
Centurion

오리톢 [902596] · MS 2019 (수정됨) · 쪽지

2024-07-03 15:18:39
조회수 576

3rd bounded cohomology of Kleinian groups - Farre(1)

게시글 주소: https://w.orbi.kr/00068609918

Theorem 1.1. Suppose $\Gamma$ is a finitely generated group that is isomorphic to a torsion-free Kleinian group without parabolics, and let $\{\rho_\alpha:\Gamma\to\mathrm{PSL}_2\Bbb C:\alpha\in\Lambda\}$ be a collection of discrete and faithful representation without parabolic or elliptic elements such that at least one of the geometrically finite end invariants of $M_{\rho_\alpha}$ is different from the geometrically infinite end invariants of $M_{\rho_\beta}$ for all $\alpha\neq \beta \in \Lambda$. Then $\{[\hat{\omega}_{\rho_\alpha}:\alpha\in\Lambda]\}$ is a linearly independent set in $H^3_b(\Gamma;\Bbb R)$.


Theorem 1.2. Suppose $\Gamma$ is a finitely generated group that is isomorphic to a torsion-free Kleinian group without parabolics. Then there is an $\epsilon = \epsilon(\Gamma)>0$ such that if $\{\rho_i:\Gamma\to\mathrm{PSL}_2\Bbb C:i = 1,\ldots, n\}$ is a collection of discrete and faithful representations without parabolics or elliptic elements such that at least one of the geometrically infinite end invariants of $M_{\rho_i}$ is different from the geometrically infinite end invariants of $M_{\rho_j}$ for all $i\neq j$ then

$$\parallel\sum_{i=1}^n a_i[\hat{\omega}_{\rho_i}]\parallel_\infty >\epsilon\max |a_i|.$$


Theorem 1.3. Let $S$ be an orientable surface with negative Euler characteristic. Then there is a constant $\epsilon = \epsilon(S)$ such that the following holds. Let $\rho:\pi_1(S)\to\mathrm{PSL}_2\Bbb C$ be discrete and faithful, without parabolic elements, and such that $M_\rho$ has at least one geometrically infinite end invariant. If $\rho':\pi_1(S)\to\mathrm{PSL}_2\Bbb C$ is any other representation satisfying

$$\parallel[\hat{\omega}_\rho] - [\hat{\omega}_{\rho'}]\parallel_\infty<\epsilon,$$

then $\rho'$ is faithful.


Corollary 1.4. Let $S$ be a closed orientable surface of genus at least 2. There is a constant $\epsilon = \epsilon(S)$ such that the following holds. Suppose that $\rho:\pi_1(S)\to\mathrm{PSL}_2\Bbb C$ is discrete, faithful, without parabolics, and such that $M_\rho$ has two geometrically infinite ends. Then for any other discrete representation $\rho':\pi_1(S)\to\mathrm{PSL}_2\Bbb C$, if

$$\parallel[\hat{\omega}_\rho] - [\hat{\omega}_{\rho'}]\parallel_\infty<\epsilon,$$

then $\rho$ and $\rho'$ are conjugate.


Corollary 1.5. There is an injective map

$$\psi:\mathcal{EL}(S)\to\overline{H}^3_b(\pi_1(S);\Bbb R)$$

whose image is a linearly independent, discrete set.


0 XDK (+0)

  1. 유익한 글을 읽었다면 작성자에게 XDK를 선물하세요.

오리톢 [902596]

쪽지 보내기


  • 첫번째 댓글의 주인공이 되어보세요.
글쓰기
오르비 1327426번째 회원 가입
ddna
1,422,035 건의 게시물이
블록체인에 디지털 공증되었습니다.
901,714
497,321
22,267
574
159

오르비 캐스트

2025 수능D - 123

오르비 플레이

오르비 RARE

오르비 과외시장
Move
our corporate site
Orbi Class
on-line classes
Atom
educational books & resources
Gae9
humor and fun

© 2000-2024 Move Inc.

orbisoptimus (v17-c3c98b)