Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric

TL;DR

This paper proves that sufficiently collapsed, closed, and irreducible three-dimensional Alexandrov spaces conform to one of Thurston's eight geometries, extending prior results from Riemannian manifolds to Alexandrov spaces.

Contribution

It extends the classification of collapsed geometries from Riemannian manifolds to the broader setting of Alexandrov spaces.

Findings

Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric.

Extension of Thurston's geometrization to Alexandrov spaces.

Generalization of prior Riemannian results.

Abstract

We prove that sufficiently collapsed, closed and irreducible three-dimensional Alexandrov spaces are modeled on one of the eight three-dimensional Thurston geometries. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting.