Belyi pairs in the critical filtrations of Hurwitz spaces

Abstract

The \it critical filtration \rm of a Hurwitz space stratifies it according to the number of the critical values; the \it Belyi pairs \rm (three critical values) are considered as the 0-dimensional strata of these filtrations. The 1-dimensional strata (four critical values) are also considered, and their irreducible components correspond to \it Fried families\rm, the introduction of which is one of the central concepts of the paper. The Belyi function on the base of a Fried family is another central concept, which is introduced and illustrated by several examples of low degree. The paper suggests the study of the combinatorial structure of the 1-dimensional strata of the critical filtration of Hurwitz spaces, extending the study of the 0-dimensional strata, to which a tremendous number of papers, including the most recent ones, is devoted (counting \it dessins d'enfants\rm). In the conclusion several directions of the further research (related to the construction of the paper) are briefly indicated. They belong to combinatorics, arithmetic, geometry of moduli spaces and Teichmuller theory.