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

## TL;DR

This paper establishes lower bounds on systole and volume for non-geometric 3-manifolds based on entropy and diameter bounds, leading to local topological rigidity results and finiteness of topological types.

## Contribution

It provides the first volume and systole bounds for non-geometric 3-manifolds in terms of entropy and diameter, and proves local topological rigidity within this class.

## Key findings

- Finitely many topological types in the class with bounded entropy and diameter.
- Closed, irreducible manifolds close in parameters are diffeomorphic.
- Examples illustrating differences from geometric cases.

## 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 "\`a 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 $\mathscr{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.