Local topological rigidity of nongeometric 3–manifolds

Cerocchi, Filippo; Sambusetti, Andrea

doi:10.2140/gt.2019.23.2899

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Local topological rigidity of nongeometric $3$–manifolds

Filippo Cerocchi and Andrea Sambusetti

Geometry & Topology 23 (2019) 2899–2927

DOI: 10.2140/gt.2019.23.2899

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.

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Keywords

entropy, systole, acylindrical splittings, $3$–manifolds

Mathematical Subject Classification 2010

Primary: 20E08, 53C23, 53C24

Secondary: 57M60, 20E08

References

Publication

Received: 17 October 2017

Revised: 25 July 2018

Accepted: 1 March 2019

Published: 1 December 2019

Proposed: Jean-Pierre Otal

Seconded: Anna Wienhard, Walter Neumann

Authors

Filippo Cerocchi
Dipartimento di Scienze di Base e Applicate per l’Ingegneria Sapienza Università di Roma Rome Italy

Andrea Sambusetti
Dipartimento di Matematica Guido Castelnuovo Sapienza Università di Roma Roma Italy

Volume 23, issue 6 (2019)