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Local topological rigidity of nongeometric $3$–manifolds

Filippo Cerocchi and Andrea Sambusetti

Geometry & Topology 23 (2019) 2899–2927

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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entropy, systole, acylindrical splittings, $3$–manifolds
Mathematical Subject Classification 2010
Primary: 20E08, 53C23, 53C24
Secondary: 57M60, 20E08
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
Filippo Cerocchi
Dipartimento di Scienze di Base e Applicate per l’Ingegneria
Sapienza Università di Roma
Andrea Sambusetti
Dipartimento di Matematica Guido Castelnuovo
Sapienza Università di Roma