Simons Collaboration Algorithms and Geometry Monthly Meeting, March 2021

Tianyi Zheng
Embedded metric structures in finitely generated amenable groups

Abstract: In this talk we will examine some examples of metric structures embedded in Cayley graphs of finitely generated groups. The starting examples are wreath products (lamplighters), where the word metric is naturally related to the Traveling Salesman Problem. Building on wreath products, we will discuss examples of groups where embedded finite metric spaces tightly control the geometry of the Cayley graph and behavior of random walks. Natural questions around metric invariants and quasi-isometry rigidity arise from the study of these groups.