This is a joint work with Laurent Bessières, Michel Boileau, Sylvain Maillot and Joan Porti. Let M be a closed, orientable, irreducible, non-simply connected 3-manifold.We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locally controlled, then M is a graph manifold. This is the last step in Perelman’s proof of Thurston’s Geometrisation Conjecture.
The reference is:
L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti Collapsing irreducible 3-manifolds with nontrivial fundamental group Invent math (2010) 179: 435–460.