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Rutgers University

1. Coskey, Samuel Gregory. Descriptive aspects of torsion-free Abelian groups.

Degree: PhD, Mathematics, 2008, Rutgers University

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.

Advisors/Committee Members: Coskey, Samuel Gregory (author), Thomas, Simon (chair), Cherlin, Gregory (internal member), Weibel, Charles (internal member), Hamkins, Joel (outside member), Deloro, Adrien (outside member).

Subjects/Keywords: Torsion free Abelian groups; Abelian groups

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Coskey, S. G. (2008). Descriptive aspects of torsion-free Abelian groups. (Doctoral Dissertation). Rutgers University. Retrieved from http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17297

Chicago Manual of Style (16th Edition):

Coskey, Samuel Gregory. “Descriptive aspects of torsion-free Abelian groups.” 2008. Doctoral Dissertation, Rutgers University. Accessed December 05, 2020. http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17297.

MLA Handbook (7th Edition):

Coskey, Samuel Gregory. “Descriptive aspects of torsion-free Abelian groups.” 2008. Web. 05 Dec 2020.

Vancouver:

Coskey SG. Descriptive aspects of torsion-free Abelian groups. [Internet] [Doctoral dissertation]. Rutgers University; 2008. [cited 2020 Dec 05]. Available from: http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17297.

Council of Science Editors:

Coskey SG. Descriptive aspects of torsion-free Abelian groups. [Doctoral Dissertation]. Rutgers University; 2008. Available from: http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17297

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