#### Volume 23, issue 6 (2019)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Local topological rigidity of nongeometric $3$–manifolds

### Filippo Cerocchi and Andrea Sambusetti

Geometry & Topology 23 (2019) 2899–2927
##### 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 $D\phantom{\rule{0.3em}{0ex}}$. 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.

##### Keywords
entropy, systole, acylindrical splittings, $3$–manifolds
##### Mathematical Subject Classification 2010
Primary: 20E08, 53C23, 53C24
Secondary: 57M60, 20E08