A GAP Package for Braid Orbit Computation and Applications

Abstract

Let G be a finite group. By Riemann's Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus, many problems on algebraic curves require the computation of braid orbits. In this paper, we describe an implementation of this computation. We discuss several applications, including the classification of irreducible familiesof indecomposable rational functions with exceptional monodromy group.

Keywords:

• Braid group
• Hurwitz space
• monodromy group of a cover
• moduli space of curves