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Countable Borel quasi-orders
Descriptive
TitleInfo
Title
Countable Borel quasi-orders
Name
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Williams
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Jay
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1985-
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Jay Williams
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author
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Thomas
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Simon
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Simon Thomas
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Advisory Committee
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chair
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Cherlin
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Gregory
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Gregory Cherlin
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Advisory Committee
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internal member
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(type = family)
Weibel
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(type = given)
Charles
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Charles Weibel
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Advisory Committee
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internal member
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Apter
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Arthur
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Arthur Apter
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Advisory Committee
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outside member
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Rutgers University
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degree grantor
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Graduate School - New Brunswick
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Text
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theses
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2012
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2012-05
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2012
Place
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xx
Language
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eng
Abstract
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In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal in cite{LR05} for the study of analytic equivalence relations, we study countable Borel quasi-orders. We are largely concerned in this thesis with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We first establish that there is a universal countable Borel quasi-order, using a Feldman-Moore-type result for countable Borel quasi-orders and an argument similar to that of Dougherty, Jackson, and Kechris in cite{DJK94}. We then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees. This is used in many of the other proofs of universality. Our main result is Theorem 5.5.2, which states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal in cite{LR05}. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. Although it may have been possible to prove this only using results on countable Borel equivalence relations, the use of quasi-orders seems to be the most direct route to this result. The proof uses small cancellation theory. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order. Finally, we discuss the structure of countable Borel quasi-orders under Borel reducibility, and we present some open problems.
Subject
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Topic
Mathematics
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Title
Rutgers University Electronic Theses and Dissertations
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ETD_3912
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electronic resource
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application/pdf
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text/xml
Extent
vi, 78 p. : ill.
Note
(type = degree)
Ph.D.
Note
(type = bibliography)
Includes bibliographical references
Note
(type = statement of responsibility)
by Jay Williams
Subject
(authority = ETD-LCSH)
Topic
Borel sets
Subject
(authority = ETD-LCSH)
Topic
Descriptive set theory
Identifier
(type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000065293
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
Identifier
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doi:10.7282/T30P0XZQ
Genre
(authority = ExL-Esploro)
ETD doctoral
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Rights
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The author owns the copyright to this work.
RightsHolder
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Name
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Williams
GivenName
Jay
Role
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RightsEvent
Type
Permission or license
DateTime
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(qualifier = exact);
(point = start)
2012-04-11 19:53:29
AssociatedEntity
Name
Jay Williams
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Affiliation
Rutgers University. Graduate School - New Brunswick
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License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
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Copyright protected
Availability
Status
Open
Reason
Permission or license
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