Rigidity of locally symmetric rank one manifolds of infinite volume
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914725706596352 |
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| author | Apanasov, Boris N. |
| author_facet | Apanasov, Boris N. |
| contents | We discuss questions by Mostow \cite{Mo1}, Bers \cite{B} and Krushkal \cite{Kr1, Kr2} about uniqueness of a conformal or spherical CR structure on the sphere at infinity $\partial H_\mathbb{F}^n$ of symmetric rank one space $H_\mathbb{F}^n$ over division algebra $\mathbb{F}=\mathbb{R}\,,\mathbb{C}\,,\mathbb{H}\,,\text{or}\,\, \mathbb{O} $ compatible with the action of a discrete group $G\subset\operatorname{Isom}H_\mathbb{F}^n$. Introducing a nilpotent Sierpiński carpet with a positive Lebesgue measure in the nilpotent geometry in $\partial H_\mathbb{F}^n\setminus\{\infty\}$ and its stretching, we construct a non-rigid discrete $\mathbb{F}$-hyperbolic groups $G\subset\operatorname{Isom}H_\mathbb{F}^n$ whose non-trivial deformations are induced by $G$-equivariant homeomorphisms of the space. Here we consider two situations: either the limit set $Λ(G)$ is the whole sphere at infinity $\partial H_\mathbb{F}^n$ or restrictions of such non-trivial deformations to components of the discontinuity set $Ω(G)\subset \partial H_\mathbb{F}^n$ are given by restrictions of $\mathbb{F}$-hyperbolic isometries. In both cases the demonstrated non-rigidity is related to non-ergodic dynamics of the discrete group action on the limit set which could be the whole sphere at infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04104 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity of locally symmetric rank one manifolds of infinite volume Apanasov, Boris N. Geometric Topology Complex Variables Dynamical Systems Metric Geometry 22E40, 32Gxx, 32M15, 32Q45, 51Mxx, 53Cxx, 53C35, 57M60, 57N16, 57S30 We discuss questions by Mostow \cite{Mo1}, Bers \cite{B} and Krushkal \cite{Kr1, Kr2} about uniqueness of a conformal or spherical CR structure on the sphere at infinity $\partial H_\mathbb{F}^n$ of symmetric rank one space $H_\mathbb{F}^n$ over division algebra $\mathbb{F}=\mathbb{R}\,,\mathbb{C}\,,\mathbb{H}\,,\text{or}\,\, \mathbb{O} $ compatible with the action of a discrete group $G\subset\operatorname{Isom}H_\mathbb{F}^n$. Introducing a nilpotent Sierpiński carpet with a positive Lebesgue measure in the nilpotent geometry in $\partial H_\mathbb{F}^n\setminus\{\infty\}$ and its stretching, we construct a non-rigid discrete $\mathbb{F}$-hyperbolic groups $G\subset\operatorname{Isom}H_\mathbb{F}^n$ whose non-trivial deformations are induced by $G$-equivariant homeomorphisms of the space. Here we consider two situations: either the limit set $Λ(G)$ is the whole sphere at infinity $\partial H_\mathbb{F}^n$ or restrictions of such non-trivial deformations to components of the discontinuity set $Ω(G)\subset \partial H_\mathbb{F}^n$ are given by restrictions of $\mathbb{F}$-hyperbolic isometries. In both cases the demonstrated non-rigidity is related to non-ergodic dynamics of the discrete group action on the limit set which could be the whole sphere at infinity. |
| title | Rigidity of locally symmetric rank one manifolds of infinite volume |
| topic | Geometric Topology Complex Variables Dynamical Systems Metric Geometry 22E40, 32Gxx, 32M15, 32Q45, 51Mxx, 53Cxx, 53C35, 57M60, 57N16, 57S30 |
| url | https://arxiv.org/abs/2401.04104 |