Dynamics of Polish groups

Kwiatkowska, Aleksandra

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https://hdl.handle.net/2142/34394 Copy

Description

Title

Dynamics of Polish groups

Author(s)

Kwiatkowska, Aleksandra

Issue Date

2012-09-18T21:14:51Z

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

Solecki, Slawomir

Doctoral Committee Chair(s)

van den Dries, Lou

Committee Member(s)

• Solecki, Slawomir
• Henson, C. Ward
• Rosendal, Christian

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
• ample generics
• Boolean actions
• isometry groups

Abstract

This thesis consists of an introduction and two independent chapters. In Chapter 2, we show that the group of all homeomorphisms of the Cantor set H(2^N) has ample generics, that is, we show that for every m the diagonal conjugacy action g\cdot (h_1, h_2,...,h_m) = (gh_1g^{-1}, gh_2g^{-1},..., gh_mg^{-1}) of H(2^N) on H(2^N)^m has a comeager orbit. This answers a question of Kechris and Rosendal. We prove that the generic tuple in H(2^N)^m can be taken to be the limit of a certain projective Fraïssé family. We also give an example of a projective Fraïssé family, which has a simpler description than the one considered in the general case, and such that its limit is a homeomorphism of the Cantor set that has a comeager conjugacy class. Additionally, using the perspective of the projective Fraïssé theory, we give examples of measures on the Cantor set such that the generic measure preserving homeomorphism exists and is realized as a projective Fraïssé limit. In Chapter 3, we prove that each measure preserving Boolean action by a Polish group of isometries of a locally compact separable metric space has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove our result, we give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert’s fifth problem plays an important role in establishing this characterization. This work is joint with Sławomir Solecki. Additionally, using our characterization, we give an alternative proof of the result by Gao and Kechris saying that no continuous action by a Polish group of isometries of a locally compact separable metric space is turbulent.

Graduation Semester

2012-08

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Copyright and License Information

Copyright 2012 Aleksandra Kwiatkowska

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