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

TL;DR

This paper proves that certain II$_1$ factors derived from irreducible lattices in products of rank one simple Lie groups are prime, providing new examples and classifying their tensor product decompositions.

Contribution

It establishes the primeness of II$_1$ factors from specific lattices and describes their tensor product decompositions, including unique prime factorization results.

Findings

II$_1$ factors from these lattices are prime

Provides the first examples of prime II$_1$ factors from higher rank semisimple Lie groups

Classifies tensor product decompositions of related II$_1$ factors

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 $d\geq 2$ with $d\not\equiv 1 mod{4}$ 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 $L(\Gamma)$.