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Journal Article

Local topological rigidity of non-geometric 3-manifolds


Cerocchi,  Filippo
Max Planck Institute for Mathematics, Max Planck Society;

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Cerocchi, F., & Sambusetti, A. (2019). Local topological rigidity of non-geometric 3-manifolds. Geometry & Topology, 23(6), 2899-2927. doi:10.2140/gt.2019.23.2899.

Cite as: https://hdl.handle.net/21.11116/0000-0006-4C3D-2
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 orresponding local topological rigidy results in the class $\mathscr{M}_{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.