Geometries for Groups (Robbie Lyman)

Abstract: The Milnor–Schwarz Lemma is a powerful tool for establishing a connection between the large-scale geometry of a group equipped with a word metric on the one hand, and a geodesic metric space on the other. This connection allows group theorists to use the powerful machinery available in geodesic metric spaces to draw conclusions about groups. A key assumption in the Milnor–Schwarz lemma is proper discontinuity or metric properness, depending on how the lemma is phrased. Recent breakthrough work of Rosendal starts from a key insight into how this assumption is really used: by shifting the hypothesis from “finiteness” to “boundedness”, the lemma begins to apply to arbitrary topological groups. I want to make this exciting insight accessible to you and report on work in progress, some of which is joint with Beth Branman, George Domat, Carlos Perez Estrada and Hannah Hoganson.