Mapping Class Groups of Rational Maps

Abstract

Given a rational map f degree at least 2, it follows from work done by McMullen and Sullivan that in certain circumstances, the pure mapping class group PMCG(f) can be identified with a subgroup of the pure mapping class group of a Riemann surface. We investigate this identification and explore what types of subgroups of mapping class groups of surfaces arise in this way. We focus primarily on the case in which PMCG(f) can be viewed as a subgroup of a product of pure mapping class groups of punctured tori. A specific case of this setting --- namely, when f is a generic quadratic rational map --- was explored by Goldberg and Keen. The authors proved that for such a choice of f, PMCG(f) is an infinitely generated subgroup of the pure mapping class group of the twice-punctured torus. We prove the analogous statement in the setting of cubic polynomials, and explicitly write down a collection of generators of PMCG(f) in terms of point-pushes and a Dehn twist. We then prove a general result that is independent of the degree of the map. Specifically, we prove that for f in an open subset of rational maps of degree d, PMCG(f) is an infinitely generated subgroup of a product of pure mapping class groups of punctured tori.