The Infinite Sphere and Galois Belyi maps

TL;DR

This paper introduces a novel geometric framework linking the space of Belyi maps to an infinite-dimensional sphere, revealing a sector decomposition that encodes monodromy and Galois properties through group-theoretic data.

Contribution

It establishes a new parametrization of Belyi maps via an infinite sphere and connects spectral sectors to algebraic quotients of free groups, providing a geometric perspective on monodromy and Galois structures.

Findings

Space of Belyi maps parametrized by an infinite sphere

Decomposition of the sphere into sectors by monodromy classes

Bijection between spectral sectors and algebraic quotients for Galois Belyi maps

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.