Prime II${}_1$ factors arising from irreducible lattices in products of rank one simple Lie groups

Abstract

We prove that if $\Gamma$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II${}_1$ factor $L(\Gamma )$ is prime. In particular, we deduce that the II${}_1$ factors associated to the arithmetic groups ${\text{PSL}}_2(\mathbb{Z}[\sqrt{d}])$ and ${\text{PSL}}_2(\mathbb{Z}[S^{-1}])$ are prime, for any square-free integer $d\ge 2$ with $d\not\equiv 1mod4$ and any finite non-empty set of primes $S$. This provides the first examples of prime II${}_1$ factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of $L(\Gamma )$ for icc countable groups $\Gamma$ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that $L(\Gamma )$ is prime, unless $\Gamma$ is a product of infinite groups, in which case we prove a unique prime factorization result for $L(\Gamma )$.