Speaker: Hiroki Matui (Chiba University)
Abstract: In this mini-course, we will introduce various properties and examples of topological full groups, which are discrete groups naturally associated with etale groupoids whose unit spaces are Cantor sets. Over the past two decades, these groups have been extensively studied and are now known to exhibit a rich array of features, such as simplicity, amenability, and finite generation, making them central objects at the intersection of group theory, topology, and operator algebras. Starting from the definition of topological full groups, we will discuss basic examples arising from minimal Z-actions and AF groupoids. We will also explain the connections to groupoid homology, and the reconstruction theorem which claims the group structure of the topological full group remembers the groupoid itself. A significant part of the theory revolves around the dichotomy of ample groupoids into two classes: almost finite and purely infinite groupoids. With this distinction in mind, we will discuss structural results such as simplicity of the commutator subgroup, amenability, and finite generation of topological full groups. Depending on time constraints, we may select certain topics to discuss in more detail.
Date & Venue:
SCMS Room 102, Sep. 9 (Tuesday), 15:00-17:00
SIMIS 18F Auditorium, Sep. 10 (Wednesday), 15:00-17:00
SCMS Room 102, Sep. 12 (Friday), 15:00-17:00
About the speaker: Hiroki Matui is a professor in the Department of Mathematics and Informatics, Chiba University. His research interests include group actions on operator algebras and connections between operator algebras and topological dynamical systems. He has published multiple papers in prestigious journals such as Invent. Math., Acta Math., J. Amer. Math. Soc., Duke Math. J., etc..
English 
简体中文