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Geodesic planes in geometrically finite acylindrical $3$-manifolds

Part of:Ergodic theoryLie groups

Published online by Cambridge University Press:  04 May 2021

YVES BENOIST and
HEE OH
YVES BENOIST*
Affiliation:
CNRS, Universite Paris-Sud, IMO, Batiment 307, 91405 Orsay, France
HEE OH
Affiliation:
Mathematics Department, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, USA (e-mail: hee.oh@yale.edu)
*
e-mail: yves.benoist@u-psud.fr
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Abstract

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Let M be a geometrically finite acylindrical hyperbolic $3$-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$-manifold $M_0$, the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$. We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

Keywords

group actionshyperbolic 3-manifoldgeodesically finite groupstopological dynamics

MSC classification

Primary:37A17: Homogeneous flows
Secondary:22E40: Discrete subgroups of Lie groups

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022, pp. 514 - 553
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Anatole Katok who was so enthusiastic in sharing with us his encyclopedic knowledge and his deep insights in dynamical systems.

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