Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric

Collapsed 3-Dimensional Alexandrov Spaces: A Brief Survey

“…In the Riemannian category, a thorough analysis of collapse of Riemannian 3-manifolds was carried out by Shioya and Yamaguchi in [28,29]. More recently, Mitsuishi and Yamaguchi obtained topological classification and structure results for collapsed Alexandrov spaces of dimension 3 (see [18]), while the authors of the present survey obtained the geometrization of closed, sufficiently collapsed irreducible 3-dimensional Alexandrov spaces (see [9]), thus extending the Riemannian results to the case of Alexandrov spaces. In this note we give a brief account of these results in Alexandrov geometry in the hope of sparking the interest of the reader.…”

“…We conclude this chapter with a brief discussion of the geometrization of sufficiently collapsed closed Alexandrov 3-spaces. We refer the reader to [9] for further details.…”

“…Classification results of Alexandrov spaces with (local) circle actions (see [10,20]) and classical results on involutions on 3-manifolds (see [15,16]) also play an important role in the proof. We refer the reader to [9] for precise details, as the proof is based on a case-by-case analysis and is of a rather technical nature.…”

Three-Dimensional Alexandrov Spaces: A Survey

“…A different but related result concerning the validity of the Borel conjecture for Alexandrov 3-spaces was obtained by Bárcenas and the second named author in [6]. In their work they reinforce the condition of asphericity further with a constraint on the Hausdorff measure of the spaces with respect to their diameters, as well as with a topological condition of irreducibility, originally defined by Galaz-García, Guijarro and the second named author in [30]. A closed Alexandrov 3-space X is irreducible if every embedded 2-sphere in X bounds a 3-ball and, in the case that the set of topologically singular points of X is non-empty, it is further required that every 2-sided RP 2 bounds a cone over a real projective plane RP 2 .…”

“…Using Mitsuishi and Yamaguchi's classification of collapsing Alexandrov 3-spaces of as well as the classification of local circle actions, Guijarro and the authors obtained a geometrization result for sufficiently collapsed Alexandrov 3-spaces [30]. Roughly speaking, they showed that a closed, irreducible and sufficiently collapsed Alexandrov 3-space X is modeled in one of the eight Thurston geometries (excluding the hyperbolic geometry H 3 ).…”