{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/26363"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/26363","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Centralizers in automorphism groups","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Hill, Aaron"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Solecki, Slawomir","Henson, C. 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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.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-07-04T18:09:57Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Hill_Aaron.pdf: 273658 bytes, checksum: eb6d4755ce955352c4ecb5fdfab7c961 (MD5)","Made available in DSpace on 2011-08-26T15:33:05Z (GMT). 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Ward","van den Dries, Lou","Rosendal, Christian"],"dc:creator":["Hill, Aaron"],"dc:date":["2011-08-26T15:33:05Z","2013-08-27T10:00:25Z","2011-08"],"dc:description":["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]. 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