Simple and large equivalence relations

TL;DR

This paper constructs special ergodic equivalence relations with unique properties, explores their surjections onto countable groups, and characterizes normal subrelations, advancing the understanding of their structure and algebraic quotients.

Contribution

It introduces ergodic equivalence relations with no proper ergodic normal or finite-index subrelations and characterizes normality and quotients of subrelations.

Findings

Constructed ergodic equivalence relations with no proper ergodic normal subrelations.

Showed surjectivity of certain treeable equivalence relations onto all countable groups.

Provided a characterization of normality and algebraic description of quotients.

Abstract

We construct ergodic discrete probability measure preserving equivalence relations $\cR$ that has no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations. 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 and an algebraic description of the quotient.