Geometry MMA Seminar: A 0-1 law for circle packings of coarsely hyperbolic metric spaces and applications to cusp excursion, part II.

Speaker: Harry Bray, George Mason University

Title:  A 0-1 law for circle packings of coarsely hyperbolic metric spaces and applications to cusp excursion, part II.

Abstract:  In the second lecture, we will with continue background on geometrically finite actions on coarsely hyperbolic metric spaces. We will then prove the logarithm law in this setting, assuming the Khinchin-type theorem. If there is interest in a third lecture, we will complete the proof of the Khinchin-type theorem. The abstract from the first lecture in the series is below:

 On the cusp of the 100 year anniversary, Khinchin’s theorem implies a strong 0-1 law for the real line; namely, the set of real numbers within distance q^{-2-\epsilon} of infinitely many rationals p/q is Lebesgue measure 0 for \epsilon>0, and full measure for \epsilon=0. In these lectures, I will present an analogous result for circle packings in Gromov hyperbolic metric spaces. As an application, we prove a logarithm law; that is, we prove asymptotics for the depth in the packing of a typical geodesic.  The first talk will be background, fundamentals, and the statements of the results. In subsequent lecture(s) we will prove the results. We will use some massive black boxes, such as quasi-independence and ergodicity of the natural boundary measures associated to these spaces.  This is joint work with Giulio Tiozzo.

Time: Monday, March 25, 2024 – 10:30am-11:45am

Place: Exploratory Hall, Room 4106