Algorithmic aspects of branched coverings I. Van Kampen's Theorem for bisets

TL;DR

This paper introduces a theory of bisets to decompose topological correspondences, extending van Kampen's theorem, and applies it to analyze Thurston maps, aiming to facilitate algorithmic classification.

Contribution

It develops a framework for decomposing bisets using graphs of bisets, generalizing van Kampen's theorem, and applies this to Thurston maps for the first time.

Findings

Decomposition of bisets into graphs of bisets analogous to van Kampen's theorem.

Application of the theory to Thurston maps and Hubbard trees.

Foundation for algorithmic decision procedures in topological dynamics.

Abstract

We develop a general theory of "bisets": sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen's theorem decomposes into a graph of groups the fundamental group of a space given with a cover, we prove analogously that the biset of a correspondence decomposes into a "graph of bisets": a graph with bisets at its vertices, given with some natural maps. The "fundamental biset" of the graph of bisets recovers the original biset. We apply these results to decompose the biset of a Thurston map (a branched self-covering of the sphere whose critical points have finite orbits) into a graph of bisets. This graph closely parallels the theory of Hubbard trees. This is the first part of a series of five articles, whose main goal is to prove algorithmic decidability of combinatorial equivalence of Thurston maps.