Orbits of braid groups on cacti

One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to $S_n$ or $A_n$, where $n$ is the degree. Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on “cacti” (ordered sets of monodromy permutations). Applying these techniques, we provide a complete topological classification for the three exceptional cases mentioned above.