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 $mathscrM_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.

Local topological rigidity of non-geometric 3-manifolds / Cerocchi, Filippo; Sambusetti, Andrea. - In: GEOMETRY & TOPOLOGY. - ISSN 1465-3060. - ELETTRONICO. - (2019). [10.2140/gt.2019.23.2899]

Local topological rigidity of non-geometric 3-manifolds

Filippo Cerocchi;Andrea Sambusetti
2019

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 $mathscrM_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.
2019
Mathematics - metric geometry; mathematics - metric geometry; mathematics - differential geometry; mathematics - geometric topology; 53C23; 53C24; 57M60; 20E08
01 Pubblicazione su rivista::01a Articolo in rivista
Local topological rigidity of non-geometric 3-manifolds / Cerocchi, Filippo; Sambusetti, Andrea. - In: GEOMETRY & TOPOLOGY. - ISSN 1465-3060. - ELETTRONICO. - (2019). [10.2140/gt.2019.23.2899]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1116576
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