PROPERTY τ AND COUNTABLE BOREL EQUIVALENCE RELATIONS

Abstract

We prove Borel superrigidity results for suitably chosen actions of groups of the form SL2(ℤ[1/p1, … , 1/pt]), where {p1, …, pt} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL2(ℤ[1/q]) on the projective lines ℚp∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.