Rigidity of Kleinian groups via self-joinings: measure theoretic criterion
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| Formato: | Preprint |
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2023
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| _version_ | 1866915452233449472 |
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| author | Kim, Dongryul M. Oh, Hee |
| author_facet | Kim, Dongryul M. Oh, Hee |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03552 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigidity of Kleinian groups via self-joinings: measure theoretic criterion Kim, Dongryul M. Oh, Hee Geometric Topology Dynamical Systems Group Theory 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. |
| title | Rigidity of Kleinian groups via self-joinings: measure theoretic criterion |
| topic | Geometric Topology Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2302.03552 |