Boundary Currents of Hitchin Components
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912331654496256 |
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| author | Reid, Charles |
| author_facet | Reid, Charles |
| contents | The space of Hitchin representations of the fundamental group of a closed surface $S$ into $\text{SL}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on $n$. We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the $\text{SL}_n\mathbb{R}$ Hitchin component is a polyhedral complex of dimension at most $n-1$. For endpoints of cubic differential rays in the $\text{SL}_3\mathbb{R}$ Hitchin component, the dual space is the universal cover of $S$, equipped with an asymmetric Finsler metric which records growth rates of trace functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary Currents of Hitchin Components Reid, Charles Geometric Topology Differential Geometry 51N15, 14M35, 57K20 The space of Hitchin representations of the fundamental group of a closed surface $S$ into $\text{SL}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on $n$. We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the $\text{SL}_n\mathbb{R}$ Hitchin component is a polyhedral complex of dimension at most $n-1$. For endpoints of cubic differential rays in the $\text{SL}_3\mathbb{R}$ Hitchin component, the dual space is the universal cover of $S$, equipped with an asymmetric Finsler metric which records growth rates of trace functions. |
| title | Boundary Currents of Hitchin Components |
| topic | Geometric Topology Differential Geometry 51N15, 14M35, 57K20 |
| url | https://arxiv.org/abs/2504.12294 |