Topological Full Groups by Example

Are there infinite, finitely generated simple groups? This seemingly innocuous question troubled some of history's best group theorists for more than 50 years. In this talk, I will explain how ample groupoids (or, through Renault duality, certain Cartan pairs) provide a unifying framework to generate and study interesting examples of these groups. This framework is called "topological full groups" and was pioneered by Hiroki Matui in the late 00's. It allowed us to answer fundamental questions like "Are there amenable, infinite, finitely generated simple groups?". I will then give some concrete examples of these "topological full groups" which I found interesting to study. These examples come from concrete C*-algebras and groupoids that I hope might be familiar to some of the audience. I will assume little prior group theory knowledge. This talk is based on some research from my thesis, and some research that I did with Eusebio Gardella.