This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets. #R##N#We introduce a canonical "Levy" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps. #R##N#As an outcome, we prove that it is decidable when two Thurston maps are equivalent. We also show that the decompositions above are computable, both in theory and in practice.
['Laurent Bartholdi', 'Dzmitry Dudko']
Algorithmic aspects of branched coverings