The Infinite Sphere and Galois Belyi maps

Noémie C. Combe

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arXiv · 2602.11778

The Infinite Sphere and Galois Belyi maps

Noémie C. Combe

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Abstract

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.

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Noémie C. Combe. 2026-02-12. The Infinite Sphere and Galois Belyi maps. https://arxiv.org/abs/2602.11778

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