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

TL;DR

This paper develops a polynomial-time method for solving conjugacy problems in mapping class bisets of Thurston maps, extending classical theory with new invariants and algorithms, especially for geometric cases like expanding maps.

Contribution

It introduces an analogue of the Birman short exact sequence for mapping class bisets, enabling reduction of conjugacy problems and providing complete invariants for geometric Thurston maps.

Findings

Polynomial-time algorithms for conjugacy and centralizer problems.

Complete conjugacy invariants for geometric Thurston maps.

Description of bisets for (2,2,2,2)-maps as crossed products.

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 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.