Simple and large equivalence relations
Lewis Bowen, Pierre-Emmanuel Caprace
Comments: Comments welcome! with an appendix by 2nd author
Subjects: Dynamical Systems (math.DS); Operator Algebras (math.OA)
We construct ergodic discrete probability measure preserving equivalence relations $R_1,R_2,R_3$ such that $R_1$ has no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations, $R_2$ has no nontrivial finite extensions, $R_3$ has finite cost and surjects onto every countable group with ergodic kernel. Moreover we show that every treeable equivalence relation satisfying a mild ergodicity condition and cost >1 surjects onto every countable group with ergodic kernel. Lastly, we provide a simple characterization of normality for subequivalence relations.
Simple and large equivalence relationsLewis Bowen, Pierre-Emmanuel CapraceComments: Comments welcome! with an appendix by 2nd authorSubjects: Dynamical Systems (math.DS); Operator Algebras (math.OA)We construct ergodic discrete probability measure preserving equivalence relations $R_1,R_2,R_3$ such that $R_1$ has no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations, $R_2$ has no nontrivial finite extensions, $R_3$ has finite cost and surjects onto every countable group with ergodic kernel. Moreover we show that every treeable equivalence relation satisfying a mild ergodicity condition and cost >1 surjects onto every countable group with ergodic kernel. Lastly, we provide a simple characterization of normality for subequivalence relations.
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