Topological full groups

Tanner, Owen (2024) Topological full groups. PhD thesis, University of Glasgow.

Abstract

We develop our understanding of topological full groups, a way of constructing examples of infinite simple groups with finiteness properties from ample groupoids. Our results are concentrated in three main example classes. Firstly, the topological full groups of purely infinite minimal groupoids share many properties with Thompson’s group V. In studying these groups and the associated groupoids in detail, we formalise this phenomenon by relating dynamical properties to group-theoretic properties. Secondly, interval exchange groups are an important concrete example of topological full groups since many are amenable. We classify these groups through computing associated Elliot invariants. Also, we find concrete generating sets and compute the homology of these groups. Thirdly and finally, Stein’s groups were introduced by Melanie Stein in 1992 as generalisations of Thompson’s group. We show these groups are topological full groups. We then analyse Stein’s groups through this framework, showing that the (simple) derived subgroups of Stein’s groups are in many cases finitely generated. We study the homology of Stein’s groups.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the European Research Council (ERC) (No. 817597), the Deutsche Forschungsgemeinschaft (DFG) (Excellence Strategy EXC 2044 –390685587), under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure, and through SFB 1442, and by the European Research Council (ERC) (Advanced Grant 834267-AMAREC).
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: European Research Council (ERC), European Commission (EC), Deutsche Forschungsgemeinschaft (DFG)
• Supervisor's Name: Li, Professor Xin
• Date of Award: 2024
• Depositing User: Theses Team
• Unique ID: glathesis:2024-84586
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 20 Sep 2024 10:23
• Last Modified: 20 Sep 2024 10:25
• Thesis DOI: 10.5525/gla.thesis.84586
• URI: https://theses.gla.ac.uk/id/eprint/84586