# Algorithmic aspects of branched coverings I/V. Van Kampen’s theorem for bisets

### Laurent Bartholdi

École Normale Supérieure, Paris, France and Universität Göttingen, Germany### Dzmitry Dudko

Jacobs-Universität Bremen, Germany

## 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 selfcovering 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.

## Cite this article

Laurent Bartholdi, Dzmitry Dudko, Algorithmic aspects of branched coverings I/V. Van Kampen’s theorem for bisets. Groups Geom. Dyn. 12 (2018), no. 1, pp. 121–172

DOI 10.4171/GGD/441