论文标题
克莱尼人团体通过自我加入的刚性
Rigidity of Kleinian groups via self-joinings
论文作者
论文摘要
令$γ<\ mathrm {psl} _2（\ Mathbb {c}）\ simeq \ simeq \ Mathrm {isom}^+（\ Mathbb {h}^3）$是有限生成的非fuch量生成的非fuch量生成的kleinian组，其普通集$ω= \ mathbb {令$ρ：γ\ to \ mathrm {psl} _2（\ mathbb {c}）$为忠实的离散的非英寸式代表，带有边界图$ f：λ\ to \ mathbb {s s}^2 $在极限集上。 在本文中，我们获得了一种新的刚性定理：如果$ f $是{\ it $λ$}的综合性，从某种意义上说，$ f $ ap $ f $映射到圆圈中的每一个$λ$的每个圆形切片，则$ f $将扩展到möbius$ g $ g $ g $ g $ $ g $ on $ \ nathbb {s}^2 $ and $ $ $ g $ g $ g $ g g $ g $ g $ g $ g $ g。此外，除非$ρ$是一种偶像，否则一组圆圈$ c $，以使$ f（c \capλ）$包含在一个圆圈中，在所有圆圈的空间中都有空的内部装饰，满足$λ$。这回答了麦克穆伦（McMullen）对地图的刚度$λ\ to \ mathbb {s}^2 $发送的一个问题，将每个四面体的每个四面体的顶点发送到零元素的四面体的顶点。 我们的证明的新颖性是将$γ$与自动键合$（\ Mathrm {id} \ timesρ）（γ）<\ Mathrm {psl} _2（\ Mathbb {C}）\ times \ times \ Mathrm {pslm {pslm {psl} _2（c）（c \ nath）（c \ nath）$ {pslm {pslm {psligrm {pslrm {$ {
Let $Γ<\mathrm{PSL}_2(\mathbb{C})\simeq \mathrm{Isom}^+(\mathbb{H}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $Ω=\mathbb{S}^2-Λ$ has at least two components. Let $ρ: Γ\to \mathrm{PSL}_2(\mathbb{C})$ be a faithful discrete non-Fuchsian representation with boundary map $f:Λ\to \mathbb{S}^2$ on the limit set. In this paper, we obtain a new rigidity theorem: if $f$ is {\it conformal on $Λ$}, in the sense that $f$ maps every circular slice of $Λ$ into a circle, then $f$ extends to a Möbius transformation $g$ on $\mathbb{S}^2$ and $ρ$ is the conjugation by $g$. Moreover, unless $ρ$ is a conjugation, the set of circles $C$ such that $f(C\cap Λ)$ is contained in a circle has empty interior in the space of all circles meeting $Λ$. This answers a question asked by McMullen on the rigidity of maps $Λ\to \mathbb{S}^2$ sending vertices of every tetrahedron of zero-volume to vertices of a tetrahedron of zero-volume. The novelty of our proof is a new viewpoint of relating the rigidity of $Γ$ with the higher rank dynamics of the self-joining $(\mathrm{id} \times ρ)(Γ)<\mathrm{PSL}_2(\mathbb{C})\times \mathrm{PSL}_2(\mathbb{C})$.
