Complex Hyperbolic Elliptics Preserving Lagrangian Planes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914395389427712 |
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| author | Xu, Mengmeng Zhang, Yibo |
| author_facet | Xu, Mengmeng Zhang, Yibo |
| contents | We prove that a regular elliptic isometry $f$ of complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^2$ preserves a Lagrangian plane through its fixed point as a non-involution if and only if $f$ is real elliptic. In this case, the isometry $f$ actually preserves a continuous one-parameter family of Lagrangian planes through the fixed point. The boundaries of these planes form a torus $\mathbb{T}^2_f \subset \partial \mathbf{H}_{\mathbb{C}}^2$, called the fixed torus of $f$. For torsion $f$, we show that all Ford domains of $\langle f \rangle$ with respect to the extended Cygan metric and centred on $\mathbb{T}^2_f$ admit the same explicit cellular structure. As an application, we classify all discrete and faithful complex hyperbolic $(n,\infty,\infty)$-triangle groups for $n = 3, 4, 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14470 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex Hyperbolic Elliptics Preserving Lagrangian Planes Xu, Mengmeng Zhang, Yibo Geometric Topology 57 M 60, 22 F 30, 22 E 40 We prove that a regular elliptic isometry $f$ of complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^2$ preserves a Lagrangian plane through its fixed point as a non-involution if and only if $f$ is real elliptic. In this case, the isometry $f$ actually preserves a continuous one-parameter family of Lagrangian planes through the fixed point. The boundaries of these planes form a torus $\mathbb{T}^2_f \subset \partial \mathbf{H}_{\mathbb{C}}^2$, called the fixed torus of $f$. For torsion $f$, we show that all Ford domains of $\langle f \rangle$ with respect to the extended Cygan metric and centred on $\mathbb{T}^2_f$ admit the same explicit cellular structure. As an application, we classify all discrete and faithful complex hyperbolic $(n,\infty,\infty)$-triangle groups for $n = 3, 4, 5$. |
| title | Complex Hyperbolic Elliptics Preserving Lagrangian Planes |
| topic | Geometric Topology 57 M 60, 22 F 30, 22 E 40 |
| url | https://arxiv.org/abs/2603.14470 |