Describing generic elements of Polish groups

Ihli, Dakota Thor

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

Description

Title

Describing generic elements of Polish groups

Author(s)

Ihli, Dakota Thor

Issue Date

2022-12-02

Director of Research (if dissertation) or Advisor (if thesis)

Tserunyan, Anush

Doctoral Committee Chair(s)

Solecki, Sławomir

Committee Member(s)

• Albin, Pierre
• Oikhberg, Timur

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Polish groups
• generic elements
• conjugacy classes
• automorphism groups
• random poset
• absolutely continuous
• topological rank

Abstract

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 is dense.

Graduation Semester

2022-12

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Dakota Thor Ihli

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