Strong primeness for equivalence relations arising from Zariski dense subgroups

TL;DR

This paper proves that certain complex orbit equivalence relations from Zariski dense subgroups are strongly prime, leading to unique prime factorizations for relations from higher rank lattice actions, extending Zimmer's results.

Contribution

It establishes strong primeness for orbit equivalence relations from Zariski dense subgroups and proves unique prime factorizations for relations from higher rank lattices.

Findings

Orbit equivalence relations from Zariski dense subgroups are strongly prime.

Unique prime factorization exists for relations from higher rank lattice actions.

Extends Zimmer's primeness results to broader classes of algebraic group actions.

Abstract

We show that orbit equivalence relations arising from essentially free ergodic probability measure preserving actions of Zariski dense discrete subgroups of simple algebraic groups are strongly prime. As a consequence, we prove the existence and the uniqueness of a prime factorization for orbit equivalence relations arising from direct products of higher rank lattices. This extends and strengthens Zimmer's primeness result for equivalence relations arising from actions of lattices in simple Lie groups. The proof of our main result relies on a combination of ergodic theory of algebraic group actions and Popa's intertwining theory for equivalence relations.