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Geodesic planes in hyperbolic 3-manifolds of infinite volume

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Speaker: 
Yongquan Zhang
Affiliation: 
MPIM
Date: 
Thu, 2021-11-04 16:30 - 17:30

Meeting-ID: 973 8400 7043
For password please contact Stephan Stadler (stadler@mpim-bonn.mpg.de).

A geodesic plane in a complete hyperbolic 3-manifold is an isometrically immersed, totally geodesic hyperbolic plane. By the works of Ratner and Shah, if the 3-manifold has finite volume, any geodesic plane is either closed or dense. Moreover, any infinite sequence of closed geodesic planes becomes densely distributed in the 3-manifold. Recent works of McMullen-Mohammadi-Oh and Benoist-Oh have generalized these properties to certain hyperbolic 3-manifolds of infinite volume, if we restrict to the interior of the convex core. In this talk, I will give a survey of these recent results, and discuss some examples where these properties fail -- for example, without restriction to the convex core, a geodesic plane can be neither closed nor dense. These examples help us to delineate the "boundary" beyond which Ratner's theorems break down.

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