Centralizers in automorphism groups

Hill, Aaron

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https://hdl.handle.net/2142/26363 Copy

Description

Title

Centralizers in automorphism groups

Author(s)

Hill, Aaron

Issue Date

2011-08-26T15:33:05Z

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

Solecki, Slawomir

Doctoral Committee Chair(s)

Henson, C. Ward

Committee Member(s)

• Solecki, Slawomir
• van den Dries, Lou
• 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)

• 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.

Graduation Semester

2011-08

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http://hdl.handle.net/2142/26363

Copyright and License Information

Copyright 2011 Aaron Hill

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