Centralizers in automorphism groups

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Hill, Aaron

Contributors dc:contributor

• Solecki, Slawomir
• Henson, C. Ward
• van den Dries, Lou
• Rosendal, Christian

• Rank-1
• Topological Complexity
• Centralizer

Statement dc:rights

• Copyright 2011 Aaron Hill

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/26363