A non-trivial family of trivial bundles with complex hyperbolic structure
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| Format: | Preprint |
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2024
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| author | Botós, Hugo C. Franco, Felipe A. |
| author_facet | Botós, Hugo C. Franco, Felipe A. |
| contents | In $\mathrm{PU}(2,1)$, the group of holomorphic isometries of the complex hyperbolic plane, we study the space of involutions $R_1, R_2, R_3, R_4, R_5$ satisfying $R_5R_4R_3R_2R_1=1$, where $R_1$ is a reflection in a complex geodesic and the other $R_i$'s are reflections in points of the complex hyperbolic plane. We show that this space modulo $\mathrm{PU}(2,1)$-conjugation is bending-connected and has dimension $4$. Using this, we construct a $4$-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle with vanishing Euler number over the sphere with $5$ cone points of angle $π$. Bending-connectedness here means that we can naturally deform the geometric structure, like Dehn twists in Teichmüller theory. Additionally, finding complex hyperbolic disc orbibundles with vanishing Euler number is a hard problem, originally conjectured by W. Goldman and Y. Eliashberg and solved by S. Anan'in and N. Gusevskii, and we produce a simpler and more straightforward construction for them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_02213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A non-trivial family of trivial bundles with complex hyperbolic structure Botós, Hugo C. Franco, Felipe A. Geometric Topology Differential Geometry 57S30 (Primary) 51M10, 57M50, 57R18 (Secondary) In $\mathrm{PU}(2,1)$, the group of holomorphic isometries of the complex hyperbolic plane, we study the space of involutions $R_1, R_2, R_3, R_4, R_5$ satisfying $R_5R_4R_3R_2R_1=1$, where $R_1$ is a reflection in a complex geodesic and the other $R_i$'s are reflections in points of the complex hyperbolic plane. We show that this space modulo $\mathrm{PU}(2,1)$-conjugation is bending-connected and has dimension $4$. Using this, we construct a $4$-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle with vanishing Euler number over the sphere with $5$ cone points of angle $π$. Bending-connectedness here means that we can naturally deform the geometric structure, like Dehn twists in Teichmüller theory. Additionally, finding complex hyperbolic disc orbibundles with vanishing Euler number is a hard problem, originally conjectured by W. Goldman and Y. Eliashberg and solved by S. Anan'in and N. Gusevskii, and we produce a simpler and more straightforward construction for them. |
| title | A non-trivial family of trivial bundles with complex hyperbolic structure |
| topic | Geometric Topology Differential Geometry 57S30 (Primary) 51M10, 57M50, 57R18 (Secondary) |
| url | https://arxiv.org/abs/2411.02213 |