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

Daniel Drimbe, Daniel J. Hoff+1 more|arXiv (Cornell University)|Nov 7, 2016

Advanced Operator Algebra Research37 references5 citations

TL;DR

This paper establishes the first examples of prime II₁ factors arising from lattices in higher rank semisimple Lie groups by proving that the group von Neumann algebra $L(\Gamma)$ is prime for any icc irreducible lattice $\Gamma$ in a product of connected non-compact rank one simple Lie groups with finite center. The result relies on measure equivalence and deformation/rigidity techniques, extending primeness results beyond rank one groups to higher rank settings.

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$_1$ factor $L(Γ)$ is prime. In particular, we deduce that the II$_1$ factors associated to the arithmetic groups $ ext{PSL}_2(\mathbb Z[\sqrt{d}])$ and $ ext{PSL}_2(\mathbb Z[S^{-1}])$ are prime, for any square-free integer $d\geq 2$ with $d ot\equiv 1 mod{4}$ 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 $L(Γ)$ for icc countable groups $Γ$ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that $L(Γ)$ is prime, unless $Γ$ is a product of infinite groups, in which case we prove a unique prime factorization result for $L(Γ)$.

Motivation & Objective

• To resolve the longstanding open problem of whether II₁ factors arising from lattices in higher rank semisimple Lie groups can be prime.
• To extend primeness results—previously limited to rank one groups like free groups or $\mathrm{SL}_2(\mathbb{R})$—to higher rank lattices.
• To establish a unique prime factorization theorem for $L(\Gamma)$ when $\Gamma$ is measure equivalent to a product of non-elementary hyperbolic groups.
• To provide the first examples of separable prime II₁ factors from lattices in higher rank Lie groups, such as $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{d}])$ and $\mathrm{PSL}_2(\mathbb{Z}[S^{-1}])$.
• To characterize all tensor product decompositions of $L(\Gamma)$ for such groups, showing that primeness holds unless $\Gamma$ is a product of infinite groups.

Proposed method

• Use of measure equivalence to relate $\Gamma$ to products of non-elementary hyperbolic groups, enabling the application of rigidity techniques.
• Application of deformation/rigidity theory and closable derivations to analyze the structure of $L(\Gamma)$, particularly its relative commutants.
• Employment of the notion of weak containment and strong asymptotic orthogonality in the context of group actions on measure spaces.
• Proof by contradiction to rule out the existence of non-trivial normal compact subgroups in the ambient group $G$, which would contradict irreducibility.
• Use of unitary conjugacy and amplification to classify tensor product decompositions of $L(\Gamma)$, showing that any decomposition must arise from a group product decomposition.
• Leveraging the fact that $L(\Gamma_i)$ is prime for each factor $\Gamma_i$ in a product decomposition to establish uniqueness of the factorization.

Experimental results

Research questions

• RQ1 Can II₁ factors arising from lattices in higher rank semisimple Lie groups be prime?
• RQ2 What are the possible tensor product decompositions of $L(\Gamma)$ for an icc group $\Gamma$ that is measure equivalent to a product of non-elementary hyperbolic groups?
• RQ3 Is there a unique prime factorization for $L(\Gamma)$ when $\Gamma$ is an irreducible lattice in a product of rank one Lie groups?
• RQ4 Do arithmetic groups such as $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{d}])$ and $\mathrm{PSL}_2(\mathbb{Z}[S^{-1}])$ give rise to prime II₁ factors?
• RQ5 What structural constraints does the existence of a non-trivial tensor decomposition impose on the group $\Gamma$?

Key findings

• The II₁ factor $L(\Gamma)$ is prime for any icc irreducible lattice $\Gamma$ in a product of connected non-compact rank one simple Lie groups with finite center.
• The arithmetic groups $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{d}])$ and $\mathrm{PSL}_2(\mathbb{Z}[S^{-1}])$ are shown to yield prime II₁ factors for any square-free $d \geq 2$ with $d \not\equiv 1 \pmod{4}$ and any finite non-empty set of primes $S$.
• For any $\Gamma \in \mathscr{L}$, the II₁ factor $L(\Gamma)$ is prime, establishing a broad class of new examples.
• If $\Gamma$ is a product of infinite groups, $L(\Gamma)$ admits a unique prime factorization into factors $L(\Gamma_i)$, each of which is prime.
• Any tensor product decomposition of $L(\Gamma)$ arises from a group product decomposition $\Gamma = \Sigma_1 \times \Sigma_2$, up to unitary conjugacy and amplification.
• The uniqueness of the decomposition implies that if $L(\Gamma) = P_1 \overline{\otimes} \cdots \overline{\otimes} P_m$, then each $P_j$ is unitarily conjugate to a tensor product of $L(\Gamma_i)$'s, and if each $P_j$ is prime, then $m = k$ and the decomposition is unique up to permutation and unitary conjugacy.