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Uniform TitleDescriptive aspects of torsion-free Abelian groups
NameCoskey, Samuel Gregory (author), Thomas, Simon (chair), Cherlin, Gregory (internal member), Weibel, Charles (internal member), Hamkins, Joel (outside member), Deloro, Adrien (outside member), Rutgers University, Graduate School - New Brunswick,
Degree Date2008-05
Date Created2008
SubjectMathematics,
Torsion free Abelian groups,
Abelian groups
DescriptionIn 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.
NotePh.D.
NoteIncludes bibliographical references (p. 73-74).
Genretheses
Persistent URLhttp://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17297
LanguageEnglish
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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