Geometrization of postcritically finite branched coverings

TL;DR

This paper investigates the canonical decompositions of postcritically finite branched coverings of the 2-sphere, demonstrating that hyperbolic cycles in these decompositions lack Thurston obstructions and are equivalent to rational maps.

Contribution

It proves that all hyperbolic cycles in the canonical decompositions are free of Thurston obstructions, establishing their Thurston equivalence to rational maps.

Findings

Hyperbolic cycles lack Thurston obstructions.

Hyperbolic cycles are Thurston equivalent to rational maps.

Canonical decompositions facilitate understanding of branched coverings.

Abstract

We study canonical decompositions of postcritically finite branched coverings of the 2-sphere, as defined by K.~Pilgrim. We show that every hyperbolic cycle in the decomposition does not have a Thurston obstruction. It is thus Thurston equivalent to a rational map.