Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence

Let f^͂: (S², A͂) ↺ be a Thurston map and let M(f^͂) be its mapping class biset: isotopy classes rel A^͂ of maps obtained by pre- and post-composing f^͂ by the mapping class group of (S², A͂). Let A ⊆ A^͂ be an f^͂-invariant subset, and let f : (S², A) ↺ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group Mod(S², A͂) is an iterated extension of Mod(S², A) by fundamental groups of punctured spheres, M(f^͂) is an iterated extension of M(f) by the dynamical biset of f. Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in M(f^͂) to that in M(f). In case f͂ is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset B(f) together with a “portrait of bisets” induced by A͂ is a complete conjugacy invariant of f^͂. Along the way, we give a complete description of bisets of (2, 2, 2, 2)-maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order 2, and we show that non-cyclic orbisphere bisets have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps.