The Topology of locally volume collapsed 3-Orbifolds

Faessler, Daniel (2011): The Topology of locally volume collapsed 3-Orbifolds. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics

Abstract

In this thesis we study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. Our main result is that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds satisfies Thurston’s Geometrization Conjecture. We also prove a version of this result with boundary. Kleiner and Lott indepedently and simultanously proved similar results ([KL11]). The main step of our proof is to construct a graph decomposition of sufficiently collapsed (closed) 3-orbifolds. We describe a coarse stratification of roughly 2-dimensional Alexandrov spaces which we then promote to a decomposition into suborbifolds for collapsed 3-orbifolds; this decomposition can then be reduced to a graph decomposition. We complete our proof by showing that graph orbifolds without bad 2-suborbifolds satisfy the Geometrization Conjecture.

• Item Type: Theses (Dissertation, LMU Munich)
• Subjects: 500 Natural sciences and mathematics > 510 Mathematics; 500 Natural sciences and mathematics
• Faculties: Faculty of Mathematics, Computer Science and Statistics
• Language: English
• Date of oral examination: 30. June 2011
• 1. Referee: Leeb, Bernhard
• Signature of the printed copy: 0001/UMC 19556