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Descriptive aspects of torsion-free Abelian groups
Descriptive
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Title
Descriptive aspects of torsion-free Abelian groups
Name
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Coskey
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Samuel Gregory
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Samuel Gregory Coskey
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author
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Thomas
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Simon
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Advisory Committee
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Simon Thomas
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chair
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Cherlin
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Gregory
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Advisory Committee
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Gregory Cherlin
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internal member
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Weibel
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Charles
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Advisory Committee
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Charles Weibel
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Hamkins
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Joel
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Advisory Committee
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Joel Hamkins
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outside member
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Deloro
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Adrien
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Advisory Committee
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Adrien Deloro
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Rutgers University
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Graduate School - New Brunswick
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theses
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2008
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2008-05
Language
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English
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electronic
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v, 75 pages
Abstract
In recent years, a major theme in descriptive set theory has been the study of the Borel complexity of naturally occurring classification
problems. For example, Hjorth and Thomas have shown that the Borel complexity of the isomorphism problem for the torsion-free abelian groups of rank n increases strictly with the rank n.
In this thesis, we present some new applications of the theory of countable Borel equivalence relations to various classification
problems for the p-local torsion-free abelian groups of finite rank. Our main result is that when ngeq3, the isomorphism and
quasi-isomorphism problems for the p-local torsion-free abelian groups of rank n have incomparable Borel complexities. (Here two
abelian groups A and B are said to be quasi-isomorphic if A is abstractly commensurable with B.) We also introduce a new invariant, the divisible rank, for the class of p-local torsion-free
abelian groups of finite rank; and we prove that if ngeq3 and 1leq kleq n-1, then the isomorphism problems for the p-local torsion-free abelian groups of rank n and divisible rank k have
incomparable Borel complexities as k varies.
Our proofs rely on the framework developed by Adams and Kechris, whereby cocycle superrigidity results from measurable group theory are applied in the purely Borel setting. In particular, we make use of the recent cocycle superrigidity theorem, due to Ioana, for free
ergodic profinite actions of Kazhdan groups.
problems. For example, Hjorth and Thomas have shown that the Borel complexity of the isomorphism problem for the torsion-free abelian groups of rank n increases strictly with the rank n.
In this thesis, we present some new applications of the theory of countable Borel equivalence relations to various classification
problems for the p-local torsion-free abelian groups of finite rank. Our main result is that when ngeq3, the isomorphism and
quasi-isomorphism problems for the p-local torsion-free abelian groups of rank n have incomparable Borel complexities. (Here two
abelian groups A and B are said to be quasi-isomorphic if A is abstractly commensurable with B.) We also introduce a new invariant, the divisible rank, for the class of p-local torsion-free
abelian groups of finite rank; and we prove that if ngeq3 and 1leq kleq n-1, then the isomorphism problems for the p-local torsion-free abelian groups of rank n and divisible rank k have
incomparable Borel complexities as k varies.
Our proofs rely on the framework developed by Adams and Kechris, whereby cocycle superrigidity results from measurable group theory are applied in the purely Borel setting. In particular, we make use of the recent cocycle superrigidity theorem, due to Ioana, for free
ergodic profinite actions of Kazhdan groups.
Note
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Ph.D.
Note
(type = bibliography)
Includes bibliographical references (p. 73-74).
Subject
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Topic
Mathematics
Subject
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Topic
Torsion free Abelian groups
Subject
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Topic
Abelian groups
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17297
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ETD_937
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NjNbRU
Identifier
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doi:10.7282/T3NS0V7D
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ETD doctoral
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Open
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Samuel Coskey
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Rutgers University. Graduate School - New Brunswick
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Non-exclusive ETD license
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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.
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