Minicourse abstracts

Inhyeok Choi (KAIS)

Title: Hyperbolicity of geodesics, counting and random walks

Abstract: In this mini-course, I aim to explain the idea behind my work with Kunal Chawla and Giulio Tiozzo, which is motivated by the work of Wenyuan Yang. There has been lots of effort to generalize classical settings of negatively curved manifolds and trees. It turns out that space possessing `hyperbolic’ directions share many behavior with negatively curved manifolds. I will first explain the contracting property of geodesics and connect them to classical settings. Then I will introduce two topics-orbit counting and random walks, and answer whether contracting property is predominant in a given space.

Emily Stark (Wesleyan)

Title: Boundaries of hyperbolic groups and their metrics

Abstract: Hyperbolic groups form a vast and beautiful family at the heart of geometric group theory. The boundary of a hyperbolic group captures the shape of the group at infinity. The boundary is a topological compactification that carries a family of visual metrics, well-defined up to quasi-symmetry. This minicourse will focus on these metrics and their structure. This analytic point of view brings powerful tools to study quasi-isometries and rigidity problems. We will present both classic results and newer applications, including joint work with Margolis—Shepherd —Woodhouse and Field—Gupta—Lyman. 

Abdul Zalloum (Toronto)

Title: Constructing hyperbolic and injective spaces via walls

Abstract: Injective metric spaces are geodesic spaces where each collection of pair-wise intersecting balls admit a total intersection. Recent work of Haettel—Hoda—Petyt shows that mapping class groups (and more generally, HHGs) admit proper cobounded actions on injective spaces leading to many interesting consequences regarding mapping class groups that were previously unknown, for instance, regarding semi-hyperbolicity and growth-related questions.


The primary example of injective spaces are CAT(0) cube complexes; these comprise a particularly well-behaved class of non-positively curved spaces and their study has led to groundbreaking advances in 3-manifold theory. Celebrated work of Sageev shows that every CAT(0) cube complex arises as a set with a collection of bi-partitions (called walls) satisfying some consistency conditions.


I will discuss CAT(0) cube complexes plus with the Sageev's construction, injective spaces along with their recent impact on the field, and finally, I will talk about recent work with Petyt and Spriano where we extend the Sageev's construction allowing it to produce actions not only on CAT(0) cube complexes, but also on injective and hyperbolic spaces starting with a set and a collection of walls satisfying some conditions. Many applications of the aforementioned extension of Sageev's construction will be discussed.