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

Let $\tilde{f}\text{\hspace{0.17em}⁣}:(S^2,\tilde{A})\to (S^2,\tilde{A})$ be a Thurston map and let $M(\tilde{f})$ be its mapping class biset: isotopy classes rel $\tilde{A}$ of maps obtained by pre- and post-composing $\tilde{f}$ by the mapping class group of $(S^2,\tilde{A})$. Let $A\subseteq \tilde{A}$ be an $\tilde{f}$-invariant subset, and let $f\text{\hspace{0.17em}⁣}:(S^2,A)\to (S^2,A)$ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group $\mathrm{M}\mathrm{o}\mathrm{d}(S^2,\tilde{A})$ is an iterated extension of $\mathrm{M}\mathrm{o}\mathrm{d}(S^2,A)$ by fundamental groups of punctured spheres, $M(\tilde{f})$ 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 $M(\tilde{f})$ to that in $M(f)$. In case $\tilde{f}$ 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 $\tilde{A}$ is a complete conjugacy invariant of $\tilde{f}$. 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.