Local topological rigidity of nongeometric $3$–manifolds

Abstract

We study Riemannian metrics on compact, orientable, nongeometric $3$–manifolds (ie those whose interior does not support any of the eight model geometries) with torsionless fundamental group and (possibly empty) nonspherical boundary. We prove a lower bound “à la Margulis” for the systole and a volume estimate for these manifolds, only in terms of upper bounds on the entropy and diameter. We then deduce corresponding local topological rigidity results for manifolds in this class whose entropy and diameter are bounded respectively by $E$ and $D$. 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 $E$ and $D$) are diffeomorphic. Several examples and counterexamples are produced to stress the differences with the geometric case.