Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

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.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3050