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Title:Dynamics of Polish groups
Author(s):Kwiatkowska, Aleksandra
Director of Research:Solecki, Slawomir
Doctoral Committee Chair(s):van den Dries, Lou
Doctoral Committee Member(s):Solecki, Slawomir; Henson, C. Ward; Rosendal, Christian
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(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.
Issue Date:2012-09-18
Rights Information:Copyright 2012 Aleksandra Kwiatkowska
Date Available in IDEALS:2012-09-18
Date Deposited:2012-08

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