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Borel superrigidity for actions of low rank lattices
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Text
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Title
Borel superrigidity for actions of low rank lattices
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ETD_2192
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000051911
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eng
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theses
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Topic
Mathematics
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(authority = ETD-LCSH)
Topic
Borel sets
Subject
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(authority = ETD-LCSH)
Topic
Descriptive set theory
Subject
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(authority = ETD-LCSH)
Topic
Lattice theory
Abstract
A major recent theme in Descriptive Set Theory has been the study of countable Borel equivalence relations on standard Borel spaces, including their structure under the partial ordering of Borel reducibility. We shall contribute to this study by proving Borel incomparability results for the orbit equivalence relations arising from Bernoulli, profinite, and linear actions of certain subgroups of $PSL_2(mathbb R)$. We employ the techniques and general strategy pioneered by Adams and Kechris in cite{AK}, and develop purely Borel versions of cocycle superrigidity results arising in the dynamical theory of semisimple groups.
Specifically, using Zimmer's cocycle superrigidity theorems cite{zim}, we will prove Borel superrigidity results for suitably chosen actions of groups of the form $PSL_2(mathcal{O})$, where $mathcal{O}$ is the ring of integers inside a multi-quadratic number field. In particular, for suitable primes $pne q$, we prove that the orbit equivalence relations arising from the natural actions of $PSL_2(mathbb Z[sqrt{q}])$ on the $p$-adic projective lines are incomparable with respect to Borel reducibility as $p$, $q$ vary. Furthermore, we also obtain Borel non-reducibility results for orbit equivalence relations arising from Bernoulli actions of the groups $PSL_2(mathcal{O})$. In particular, we show that if $E_p$ denotes the orbit equivalence relation arising from a nontrivial Bernoulli action of $PSL_2(mathbb Z[sqrt{p},])$, then $E_p$ and $E_q$ are incomparable with respect to Borel reducibility whenever $pne q$.
Specifically, using Zimmer's cocycle superrigidity theorems cite{zim}, we will prove Borel superrigidity results for suitably chosen actions of groups of the form $PSL_2(mathcal{O})$, where $mathcal{O}$ is the ring of integers inside a multi-quadratic number field. In particular, for suitable primes $pne q$, we prove that the orbit equivalence relations arising from the natural actions of $PSL_2(mathbb Z[sqrt{q}])$ on the $p$-adic projective lines are incomparable with respect to Borel reducibility as $p$, $q$ vary. Furthermore, we also obtain Borel non-reducibility results for orbit equivalence relations arising from Bernoulli actions of the groups $PSL_2(mathcal{O})$. In particular, we show that if $E_p$ denotes the orbit equivalence relation arising from a nontrivial Bernoulli action of $PSL_2(mathbb Z[sqrt{p},])$, then $E_p$ and $E_q$ are incomparable with respect to Borel reducibility whenever $pne q$.
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electronic resource
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vi, 108 p.
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Ph.D.
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Includes bibliographical references (p. 104-107)
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by Scott Schneider
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Schneider
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Scott
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1980-
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Scott Schneider
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Thomas
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Simon
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chair
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Simon Thomas
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Cherlin
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Gregory
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Gregory Cherlin
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Weibel
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Charles
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Charles Weibel
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Hamkins
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Joel
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Joel Hamkins
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Rutgers University
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Graduate School - New Brunswick
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2009
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2009-10
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xx
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Rutgers University Electronic Theses and Dissertations
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ETD
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T37M0834
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ETD doctoral
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Rights
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The author owns the copyright to this work
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Copyright protected
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Open
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Mandated by sponsor
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Schneider
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Scott
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Scott Schneider
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Rutgers University. Graduate School - New Brunswick
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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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