Abelian Group Actions and Hypersmooth Equivalence Relations

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We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

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Cotton, Michael R. May 2019.

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  • Cotton, Michael R.

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We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

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  • May 2019

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  • June 10, 2019, 9:31 a.m.

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Cotton, Michael R. Abelian Group Actions and Hypersmooth Equivalence Relations, dissertation, May 2019; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc1505289/: accessed August 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .