Harrison Bray (George Mason University)
Title: A 0-1 law for geometrically finite actions on coarsely hyperbolic metric spaces
Abstract: 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 with Tiozzo for certain geometrically finite actions on Gromov hyperbolic metric spaces. As an application, we prove a generalization of Sullivan’s logarithm law, which describes the depth of a typical geodesic in a group invariant horoball packing. Along the way, I will discuss Sullivan’s work in the classical setting of geometrically finite Kleinian groups, as well as the general theory of Tukia, Yaman, and Bowditch for geometrically finite group actions on hyperbolic metric spaces.
Introductory talk by: Madeline Horton (George Mason University)
Didac Martinez-Granado (University of Luxembourg/NUS)
Title: The space of metrics of a group: the geometric group theorist's Teichmüller space
Abstract: In geometric group theory, one typically studies whether a given group G admits particular geometric structures—such as cubulations, small actions on real trees, measured wall structures, or certain random walks—all of which give rise to (invariant) metrics on G. This mini-course adopts a broader perspective: rather than focusing on each structure in isolation, we consider *all* possible metrics on G at once and investigate the global geometry of this “space of metrics.” Introduced by Furman in the early 2000s, it has recently undergone intense study.
A motivating analogy comes from the Teichmüller space of a surface, which parametrizes all marked hyperbolic metrics and is a finite-dimensional ball. However, even when G is a surface group, its space of metrics is infinitely dimensional, incorporating diverse constructions such as hyperbolic metrics, word-metrics, geodesic currents, and cubulations. We will describe some of these metrics, focusing on geodesic currents as a motivating example. More generally, for a non-elementary hyperbolic group, one can define a natural generalization of geodesic currents, "co-geodesic currents" (MG-Reyes), which are, roughly speaking, measures on "boundary hyperplanes", and they induce metrics on a hyperbolic group. Another avenue to obtain metrics on G from measures is from certain "admissible" random walks with respect to probability measures on G. These induce Green metrics which naturally sit in this space.
We will introduce and analyze a natural distance on the space of metrics, discussing in particular how both word-metrics and those arising from admissible random walks are dense (Cantrell-MG-Reyes).
If time allows, we might briefly explore how some of these ideas extend beyond hyperbolic groups, drawing connections to questions of growth in the mapping class group.
The content of the course will be based on the recent work of Reyes, Cantrell-Tanaka, Cantrell-Reyes, MG-Reyes, Cantrell-MG-Reyes, MG-Thurston, DeRosa-MG, Jyothis-MG and Ding-MG-Zalloum among others.
Introductory speaker: Meenakshy Jyothis (Binghamton University)
Jenya Sapir (Binghamton University)
Title: Counting (short) curves on surfaces
Abstract: Our goal is to understand what a "typical" short curve on a random large genus hyperbolic surface looks like. In particular, for each $L$, there are finitely many curves of length at most $L$. We find length scales at which such a curve chosen at random is highly likely to be non-simple, or fill the whole surface.
It is known that, with respect to many commonly studied random models, a typical surface will be expander. That is, it will be "highly connected," in the sense that we get effective mixing of the geodesic flow. In the first lecture, we will give an overview of the results that hold for all expander surfaces, and hence for random surfaces with respect to many different random models. This is joint work with Ben Dozier.
The curve counting techniques of Margulis are an important ingredient in these results. Margulis developed these techniques in his thesis, to count the number of closed curves of length at most $L$, as $L$ goes to infinity, on any negatively curved surface. We will give an overview of the proof this classical result. In particular, we will see how to use the dynamics of the geodesic flow on a hyperbolic surface to count the number of closed curves.
Finally, we will discuss how to adapt these techniques to counting curves of "short" length, with or without self-intersections. By comparing these two counts, we get our main result about which curves are typical.
Introductory talk by: Aisha Mechery (Rice University)