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Centralizers in automorphism groups

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Title: Centralizers in automorphism groups
Author(s): Hill, Aaron
Director of Research: Solecki, Slawomir
Doctoral Committee Chair(s): Henson, Ward
Doctoral Committee Member(s): Solecki, Slawomir; van den Dries, Lou; Rosendal, Christian
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Rank-1 Topological Complexity Centralizer
Abstract: In this dissertation we investigate centralizers in several automorphism groups of homogenous structures. In the rst chapter, we discuss the centralizer question in ergodic theory, an open question that has served as motivation for much of the work in this dissertation. We also introduce the content of each of the subsequent chapters and describe how it relates to the centralizer question in ergodic theory. In the second chapter, we investigate the topological complexity of the set of n-th powers in the group of isometries of Baire space. We prove that for n > 1, this set is not Borel. In the third chapter, we investigate topological similarity, an equivalence re- lation on a Polish group introduced by Rosendal in [15]. We prove some results for topological similarity in general Polish groups and give some new, simpli ed proofs of known genericity results in the group of invertible measure-preserving transformations. We also show that a generic measure-preserving transformation is not conjugate to any of its n-th roots, for n > 1. In the fourth chapter, we introduce the notion of a rank-1 homeomorphism of a zero-dimensional Polish space X, analogous in many ways to a rank-1 invertible measure-preserving transformation. We show that every rank-1 homeomorphism with a non-repeating tower representation (the class of such homeomorphisms is large) has trivial centralizer in the group of homeomorphisms of X.
Issue Date: 2011-08-26
URI: http://hdl.handle.net/2142/26363
Rights Information: Copyright 2011 Aaron Hill
Date Available in IDEALS: 2013-08-27
Date Deposited: 2011-08
 

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