The Topology of locally volume collapsed 3-Orbifolds

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.