通过自联结的克莱因群刚性：测度论准则
Rigidity of Kleinian groups via self-joinings: measure theoretic criterion
AI总结：
本文通过自联结子群的遍历理论与共形测度理论，证明了扎里斯基稠密凸紧克莱因群表示的测度论刚性二分法，并将结果推广至发散型子群。
摘要：
设$n,m\ge2$。令$Γ<\text{SO}^\circ(n+1,1)$为扎里斯基稠密的凸紧子群，$Λ\subset\mathbb{S}^n$为其极限集。设$ρ:Γ\to\text{SO}^\circ(m+1,1)$为扎里斯基稠密的凸紧忠实表示，$f:Λ\to\mathbb{S}^{m}$为$ρ$-边界映射。定义$Λ_f:=\bigcup\left\{C\capΛ:\begin{matrix}C\subset\mathbb{S}^n\text{是圆且}\f(C\capΛ)\text{含于}\mathbb{S}^m\text{的真球面}\end{matrix}\right\}$。当$\mathbb{S}^n$中存在至少一个$Λ$-双稳定圆时（如$Ω=\mathbb{S}^n-Λ$不连通），我们证明以下二分法：$Λ_f=Λ$或$\mathcal{H}^δ(Λ_f)=0$，其中$\mathcal{H}^δ$是维数为$δ=\dim_HΛ$的豪斯多夫测度。此外，在前一种情况下，$n=m$且$ρ$是$\mathbb{S}^n$上莫比乌斯变换的共轭。我们的证明将方向对角流的遍历理论与高阶半单李群离散子群的共形测度理论应用于自联结子群$Γ_ρ=(\operatorname{id}\timesρ)(Γ)<\text{SO}^\circ(n+1,1)\times\text{SO}^\circ(m+1,1)$。我们还得到了任何发散型子群的类似定理。
英文摘要：
Let $n, m\ge 2$. Let $Γ<\text{SO}^\circ(n+1,1)$ be a Zariski dense convex cocompact subgroup and $Λ\subset\mathbb{S}^n$ be its limit set. Let $ρ: Γ\to \text{SO}^\circ(m+1,1)$ be a Zariski dense convex cocompact faithful representation and $f:Λ\to \mathbb{S}^{m}$ the $ρ$-boundary map. Let $$Λ_f:= \bigcup \left\{ C \cap Λ: \begin{matrix} C \subset \mathbb{S}^n \text{ is a circle such that} \\ f(C \cap Λ) \text{ is contained in a proper sphere } \text{in } \mathbb{S}^m \end{matrix} \right\}.$$ When there exists at least one $Λ$-doubly stable circle in $\mathbb{S}^n$ (e.g., $Ω=\mathbb{S}^n-Λ$ is disconnected), we prove the following dichotomy: $$\text{either}\quad Λ_f= Λ\quad \text{ or } \quad \mathcal{H}^δ(Λ_f) =0,$$ where $\mathcal{H}^δ$ is the Hausdorff measure of dimension $δ=\dim_H Λ$. Moreover, in the former case, we have $n=m$ and $ρ$ is a conjugation by a Möbius transformation on $\mathbb{S}^n$. Our proof uses ergodic theory for directional diagonal flows and conformal measure theory of discrete subgroups of higher rank semisimple Lie groups, applied to the self-joining subgroup $Γ_ρ=(\operatorname{id} \times ρ)(Γ) < \text{SO}^\circ(n+1,1)\times \text{SO}^\circ(m+1,1)$. We also obtain an analogous theorem for any divergence-type subgroup.
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