Belyi Pairs and Fried Families

Abstract

For the rational functions on algebraic curves, the number of critical values typical for the Belyi functions is three. This number is replaced by four. The emerging objects, unlike the rigid Belyi pairs, can be deformed; they constitute one-parameter families. The main properties of the corresponding curves in the moduli spaces are established, and the relation of these curves to the Belyi pairs is considered. The case of the clean Belyi pairs of genus 2 of minimal degree 8 is analyzed.