BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Minicourse: topological full groups (1) - Hiroki Matui (Chiba Uni versity) DTSTART:20250630T150000Z DTEND:20250630T163000Z UID:TALK233155@talks.cam.ac.uk DESCRIPTION: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 t he past two decades\, these groups have been extensively studied and are n ow known to exhibit a rich array of features\, such as simplicity\, amenab ility\, and finite generation\, making them central objects at the interse ction of group theory\, topology\, and operator algebras. \;Starting f rom the definition of topological full groups\, we will discuss basic exam ples arising from minimal Z-actions and AF groupoids. We will also explain the connections to groupoid homology\, and the reconstruction theorem whi ch claims the group structure of the topological full group remembers the groupoid itself.A significant part of the theory revolves around the dicho tomy of ample groupoids into two classes: almost finite and purely infinit e groupoids. With this distinction in mind\, we will discuss structural re sults such as simplicity of the commutator subgroup\, amenability\, and fi nite generation of topological full groups. Depending on time constraints\ , we may select certain topics to discuss in more detail. \;\nFuther p repartory reading (this will not be assumed)\nSection 2: Preliminaries of the paper: X. Li\, &rdquo\;Ample groupoids\, topological full groups\, alg ebraic K-theory spectra and infinite loop spaces\,&rdquo\; Forum Math. Pi 13 (2025)\, (available at arXiv:2209.08087).\nH. Matui\, &rdquo\;Topologic al full groups of etale groupoids\,&rdquo\; in Operator Algebras and Appli cations&mdash\;The Abel Symposium 2015\, Abel Symp. 12\, Springer\, 2016\, (available at arXiv:1602.00383) LOCATION:External END:VEVENT END:VCALENDAR