Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

Abstract

Abstract We prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II 1 {\mathrm{II}_{1}} factor L ⁢ ( Γ ) {L(\Gamma)} is prime. In particular, we deduce that the II 1 {\mathrm{II}_{1}} factors associated to the arithmetic groups PSL 2 ⁢ ( ℤ ⁢ [ d ] ) {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and PSL 2 ⁢ ( ℤ ⁢ [ S - 1 ] ) {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer d ≥ 2 {d\geq 2} with d ≢ 1 ⁢ ( mod ⁡ 4 ) {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime II 1 {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of L ⁢ ( Γ ) {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that L ⁢ ( Γ ) {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for L ⁢ ( Γ ) {L(\Gamma)} .