The Infinite Sphere and Galois Belyi maps
Noémie C. Combe
TL;DR
The paper builds a geometric–probabilistic framework that reinterprets the space of Belyi maps as sectors of an infinite-dimensional sphere $\mathscr{S}_\infty$ derived from Voiculescu’s free probability. By studying microstate sectors $\Gamma_m(\nu,\varepsilon)$ and their large-deviation limits, it shows that atypical spectral laws force the limiting noncommutative algebra to be a quotient $L(\mathbb{F}_2/N)$ of the free group factor, with $N$ finite-index and normal. In the Galois case, this yields a genuine bijection between sectors and monodromy quotients, refining the classical monodromy classification and refining Hurwitz-type moduli spaces; in the non-Galois setting, sectors distinguish covers sharing the same monodromy. The approach connects Belyi monodromy, noncommutative probability, and random-matrix large deviations to provide a continuous geometric organization of Belyi maps and their Hurwitz data, with consequences for arithmetic geometry via profinite completions $\widehat{\mathbb{F}_2}$ and associated von Neumann algebras.
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.