Algorithmic aspects of branched coverings II. Sphere bisets and their decompositions

We consider the action of mapping class groups, by preand postcomposition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-coverings of the sphere in terms of the conjugacy and centralizer problems in a mapping class biset, and use our decomposition results on mapping class bisets to reduce these decision problems to small mapping class bisets. This is the main step in our proof of decidability of Thurston equivalence, as outlined in the first article of the series, since decidability of conjugacy and centralizer problems in the small mapping class bisets are well understood in terms of linear algebra, group theory and complex analysis. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are sphere