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Local topological
rigidity of nongeometric $3$–manifolds
Filippo Cerocchi and Andrea Sambusetti |
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Geometry & Topology 23 (2019) 2899–2927
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Abstract |
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We study Riemannian metrics on compact, orientable, nongeometric –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 and . 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 and ) are diffeomorphic. Several examples and counterexamples are produced to stress the differences with the geometric case. |
Keywords
entropy, systole, acylindrical splittings, $3$–manifolds
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Mathematical Subject Classification 2010
Primary: 20E08, 53C23, 53C24
Secondary: 57M60, 20E08
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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
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Authors |
| Filippo Cerocchi | |
| Dipartimento di Scienze di Base e
Applicate per l’Ingegneria Sapienza Università di Roma Rome Italy |
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| Andrea Sambusetti | |
| Dipartimento di Matematica Guido
Castelnuovo Sapienza Università di Roma Roma Italy |
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