Let f~:(S2,A~)→(S2,A~) 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,A~). Let A⊆A~ be an f~-invariant subset, and let f:(S2,A)→(S2,A) be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group Mod(S2,A~) 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 f~ 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 A~ 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.
Submitted 13 February, 2018; v1 submitted 8 February, 2018; originally announced February 2018.