Weak collapsing and geometrisation of aspherical 3-manifolds

Laurent Bessières (1) , Gérard Besson (1) , Michel Boileau (2) , Sylvain Maillot (3) , Joan Porti (4)

1 IF - Institut Fourier
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 IRMA - Institut de Recherche Mathématique Avancée
4 UAB - Departament de Matemàtiques [Barcelona]

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman's proof of the Geometrisation Conjecture for aspherical 3-manifolds.

Domaines: Topologie géométrique [math.GT]

Laurent Bessières, Gérard Besson, Michel Boileau, Sylvain Maillot, Joan Porti. Weak collapsing and geometrisation of aspherical 3-manifolds. 2008. ⟨hal-00437843⟩