Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, January 31, 2022 - 12:00 to 12:50

Location:

WB 205 & Zoom (contact guoren@oregonstate)

Guest Speaker:

Institution:

Max Planck Institute for Mathematics, Bonn

Abstract:

Given the classical Apollonian gasket, what are all the circles that intersect it in countably many points, and how are they distributed? Equivalently, what are all the elementary geodesic planes in the Apollonian orbifold, and how do they behave topologically?

In this talk, I will answer this question by introducing a way to encode points on the Apollonian gasket with words in three letters, analogous to cutting sequences for real numbers with respect to the modular group PSL(2,Z). This gives a coding for closed geodesics in the Apollonian orbifold as well. The goal is then to find all the words for core geodesics of elementary planes.

If time permits, I will also talk about the broader problem of understanding the topological behavior of geodesic planes in hyperbolic 3-manifolds, including recent developments and their relations to the question at hand.