Groups Logic and Dynamics
Fall Fest 2023

Using tools from the theory of optimal transport and linear representations on Banach spaces, we show how to integrate potentially unbounded Lipschitz functions defined on metric spaces acted upon by an amenable group. This is done by first establishing a Reiter style criterion for amenability of topological groups, but formulated for the Wasserstein distance between probability measures as opposed to the usual \(L_1\) distance. Some of our results improve on previous investigations by F. M. Schneider- A. Thom and M. Cuth-M. Doucha.

Group actions on trees are ubiquitous in geometric group theory and a fruitful source of both strange examples and powerful theorems. If a group acts without global fixed point on a tree, it typically does so in many ways. Forester, Clay and Guirardel–Levitt initiated the study of deformation spaces of tree actions. The purpose of this talk is to introduce these deformation spaces. Along the way I hope to state a few of my theorems and some open questions.

We introduce a “purely Borel” version of Gromov’s notion of asymptotic dimension, and show how to use it to establish hyperfiniteness of various equivalence relations. Time permitting, we discuss how these methods yield the first known examples of finitely generated hyperfinite groups of exponential growth. This is joint work with Jackson, Marks, Seward, and Tucker-Drob.

We will discuss the notion of length functions on groups, focusing on lattices in Lie groups, and discuss how higher rank dynamics can help to understand some questions about them.

An equivalence relation is hyperfinite if it is generated by a continuous action of \(\mathbb{Z}\) on a Polish space. It is known that a countable discrete group is amenable iff each of its Borel actions induces a measure-hyperfinite equivalence relation, that is, it is hyperfinite off a set of measure 0 for any Borel probability measure. It is a long-standing open conjecture that measure-hyperfiniteness implies hyperfiniteness, and the most recent progress in this direction shows that certain classes of amenable groups (such as nilpotent groups, polycyclic groups) always induce hyperfinite equivalence relations. On the other hand, many non-amenable groups induce natural equivalence relations which turn out to be hyperfinite, and this also provides a good source of examples, and maybe even potential counterexamples to the conjecture. We give an overview of various hyperfinite actions of non-amenable groups, primarily in the context of boundary actions, and actions on Cantor space. This includes joint work with Frisch, Kunnawalkam Elayavalli, Oyakawa, Spaas and Iyer.