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.