The Infinite Sphere and Galois Belyi maps
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918333668917248 |
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| author | Combe, Noémie C. |
| author_facet | Combe, Noémie C. |
| contents | We show that the space of Belyi maps admits a natural parametrization by an infinite-dimensional sphere arising from Voiculescu's theory of noncommutative probability spaces. We show that this sphere decomposes into sectors, each of which corresponds to a class of Belyi maps distinguished up to isomorphism by their monodromy, encoded by a finite-index subgroup of F2. For Galois Belyi maps, our correspondence between spectral sectors of the infinite sphere and algebraic quotients of F2 yields a genuine bijection. Within this framework, distinct sectors of the sphere capture the algebraic constraints imposed on the monodromy, thereby providing a geometric organization of Belyi maps according to their associated group-theoretic data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11778 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Infinite Sphere and Galois Belyi maps Combe, Noémie C. Algebraic Geometry Number Theory We show that the space of Belyi maps admits a natural parametrization by an infinite-dimensional sphere arising from Voiculescu's theory of noncommutative probability spaces. We show that this sphere decomposes into sectors, each of which corresponds to a class of Belyi maps distinguished up to isomorphism by their monodromy, encoded by a finite-index subgroup of F2. For Galois Belyi maps, our correspondence between spectral sectors of the infinite sphere and algebraic quotients of F2 yields a genuine bijection. Within this framework, distinct sectors of the sphere capture the algebraic constraints imposed on the monodromy, thereby providing a geometric organization of Belyi maps according to their associated group-theoretic data. |
| title | The Infinite Sphere and Galois Belyi maps |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2602.11778 |