Abstract
Let $\tilde f\colon(S^2,\tilde A)\to(S^2,\tilde A)$ 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 $f\colon(S^2,A)\to(S^2,A)$ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group $\mathrm{Mod}(S^2,\tilde A)$ is an iterated extension of $\mathrm{Mod}(S^2,A)$ 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 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 of $(2,2,2,2)$-maps as a crossed product of of torus endomorphisms by the cyclic group of order $2$, and we show that non-cyclic orbisphere have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for of expanding or torus maps.
Highlights
SetAPTf h:“urŤstonně1mfanppcisriaticbarlanpcohinetds covering of f q is f : S2 finite.ý of the sphere whose post-criticalExtending [18], we developed in [2,3,4,5] an algebraic machinery that parallels the topological theory of Thurston maps: one considers the orbisphere pS2, Pf, ordf q, with ordf : Pf Ñ t2, 3, . . . , 8u defined by ordf ppq “ l. c. m.tdegqpf nq | n ě 0, f npqq “ pu, and the orbisphere fundamental group G “ π1pS2, Pf, ord, ̊q, which has one generator of order ordf ppq per point p P Pf, and one relation
There is an efficient algorithm that decides whether a rational map is equivalent to a given twist of itself, when only extra marked points are twisted
The state of the art is based on the Nielsen-Thurston classification: decompose maps along their canonical multicurve; a complete conjugacy invariant of the map is given by the combinatorics of the decomposition, the conjugacy classes of return maps, and rotation parameters along the multicurve
Summary
SetAPTf h:“urŤstonně1mfanppcisriaticbarlanpcohinetds covering of f q is f : S2 finite. Since geometric maps have only finitely many periodic points of given period, we obtain a good understanding of conjugacy and centralizers of geometric bisets: Theorem E (= Theorem 4.41). Given a minimal geometric orbisphere biset GBG, an extension f : Ar Ñ Ar of the dynamics of B on its peripheral classes, and two portraits of bisets pGa, BaqaPAr and pG1a, Ba1 qaPAr with dynamics f, Decide whether pGa, BaqaPAr and pG1a, Ba1 qaPAr are conjugate, and Compute the centralizer of pGa, BaqaPAr, which is a finite abelian group. There is an efficient algorithm with oracle that, given two orbisphere maps f, g by their bisets and such that f is geometric, decides whether f, g are conjugate, and computes the centralizer of f. There is an efficient algorithm that decides whether a rational map is equivalent to a given twist of itself, when only extra marked points are twisted
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