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Local topological rigidity of non-geometric $3$-manifolds. (arXiv:1705.06213v1 [math.MG])
来源于:arXiv
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 $\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
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