Algorithmic aspects of branched coverings

TL;DR

This paper develops an algorithmic framework for analyzing Thurston maps by decomposing branched coverings into simpler components, proving decidability of their equivalence, and demonstrating the computability of these decompositions.

Contribution

It introduces a canonical decomposition of Thurston maps into fundamental components and proves the decidability of their equivalence, advancing the algorithmic understanding of these maps.

Findings

Decidability of Thurston map equivalence.

Computability of map decompositions.

Canonical decomposition into simpler components.

Abstract

This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets. We introduce a canonical "Levy" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps. As an outcome, we prove that it is decidable when two Thurston maps are equivalent. We also show that the decompositions above are computable, both in theory and in practice.