Weak collapsing and geometrisation of aspherical 3-manifolds

Abstract

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.

Improved version in English

References in corpus (5)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Perelman's Stability Theorem
• Multicomplexes, bounded cohomology and additivity of simplicial volume
• Canonical metrics on 3-manifolds and 4-manifolds

Cited by in corpus (6)

• Completion of the Proof of the Geometrization Conjecture
• Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds
• A gap property for the growth of closed 3-manifold groups
• Some applications of Ricci flow to 3-manifolds
• Perelman's collapsing theorem for 3-manifolds
• Deformation of Hyperbolic Cone-Structures: Study of the non-Colapsing case