Borel superrigidity for actions of low rank lattices

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 $pne q$.%%%%