Hausdorff Dimension of non-conical and Myrberg limit sets
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915328058982400 |
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| author | Mj, Mahan Yang, Wenyuan |
| author_facet | Mj, Mahan Yang, Wenyuan |
| contents | In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of $G$ in the following cases. 1. $M$ is a finite volume complete Riemannian manifold of pinched negative curvature and $G$ is an infinite normal subgroups of infinite index in $π_1(M)$. 2. $G$ acts on a regular tree $X$ with $X/G$ infinite and amenable (dimension 1). 3. $G$ acts on the hyperbolic plane $\mathbb H^2$ such that $\mathbb H^2/G$ has Cheeger constant zero (dimension 2). 4. $G$ is a finitely generated geometrically infinite Kleinian group (dimension 3). We also show that the Hausdorff dimension of the Myrberg limit set is the same as the critical exponent, confirming a conjecture of Falk-Matsuzaki. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff Dimension of non-conical and Myrberg limit sets Mj, Mahan Yang, Wenyuan Group Theory Dynamical Systems Geometric Topology In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of $G$ in the following cases. 1. $M$ is a finite volume complete Riemannian manifold of pinched negative curvature and $G$ is an infinite normal subgroups of infinite index in $π_1(M)$. 2. $G$ acts on a regular tree $X$ with $X/G$ infinite and amenable (dimension 1). 3. $G$ acts on the hyperbolic plane $\mathbb H^2$ such that $\mathbb H^2/G$ has Cheeger constant zero (dimension 2). 4. $G$ is a finitely generated geometrically infinite Kleinian group (dimension 3). We also show that the Hausdorff dimension of the Myrberg limit set is the same as the critical exponent, confirming a conjecture of Falk-Matsuzaki. |
| title | Hausdorff Dimension of non-conical and Myrberg limit sets |
| topic | Group Theory Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2506.04955 |