Local topological rigidity of non-geometric $3$-manifolds

Abstract

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "à la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then deduce corresponding local topological rigidy results in the class ${\mathcal{M}}_{ngt}^{\partial }(E,D)$ of compact non-geometric 3-manifolds with torsionless fundamental group (with possibly empty, non-spherical boundary) whose entropy and diameter are bounded respectively by $E,D$. For instance, this class locally contains only finitely many topological types; and closed, irreducible manifolds in this class which are close enough (with respect to $E,D$) are diffeomorphic. Several examples and counter-examples are produced to stress the differences with the geometric case.

21 pages

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• On the peripheral subgroups of irreducible 3-manifold groups and acylindrical splittings

Cited by in corpus (2)

• Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces
• On the peripheral subgroups of irreducible 3-manifold groups and acylindrical splittings