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

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 III

Abstract:  In the third lecture, we will prove the logarithm law in the setting of geometrically finite actions on coarsely hyperbolic metric spaces with mixed exponential growth. We will then prove the Khinchin-type theorem assuming some number of black boxes, time permitting. 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, April 1, 2024 – 10:30am-11:45am

Place: Exploratory Hall, Room 4106