Describing generic elements of Polish groups

For a Polish group G, we say an element g ∈ G is generic if it has a comeagre conjugacy class. We examine the questions of existence, and of behavior, for generic elements in two particular Polish groups. Firstly, we give a concrete model-theoretic characterization of the generic automorphism of the countable universal ultrahomogeneous partial order, also called the random poset. Secondly, we prove that the group of order-preserving, absolutely continuous homeomorphisms of the interval admits generic elements, and we give a concrete characterization. Additionally, we show that this group is generically topologically 2-generated; that is, for a comeagre set of pairs (f,g) of absolutely continuous interval homeomorphisms, the subgroup <f,g> is dense.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Ihli, Dakota Thor

Contributors dc:contributor

• Tserunyan, Anush
• Solecki, Sławomir
• Albin, Pierre
• Oikhberg, Timur

• Polish Groups
• Generic Elements
• Conjugacy Classes
• Automorphism Groups
• Random Poset
• Absolutely Continuous
• Topological Rank

Statement dc:rights

• Copyright 2022 Dakota Thor Ihli

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/117798

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F117798