Analysis on simple Lie groups and lattices · 2022-ana-189

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. Since $\mathrm{SL}_3(\mathbb Z)$ has trivial center, it is $\mathrm{PSL}_3(\mathbb Z)$, so the assertion is a subfamily of Connes' still-open rigidity problem asking whether the factors $\mathcal L(\mathrm{PSL}_m(\mathbb Z))$ and $\mathcal L(\mathrm{PSL}_n(\mathbb Z))$ can be isomorphic only when $m=n$. No proof or counterexample covering any of the requested integers $n>4$ was found.

The strongest progress supplies a possible invariant but does not yet distinguish the factors. Vincent Lafforgue and Mikael de la Salle, Noncommutative $L^p$-spaces without the completely bounded approximation property, Duke Mathematical Journal 160 (2011), 71-116, DOI 10.1215/00127094-1443478, prove that $L_p(\mathcal L(\mathrm{SL}_3(\mathbb Z)))$ fails the operator space approximation property, and hence the completely bounded approximation property, for $p>4$ (and, by duality, for $p<4/3$). This gives no separation in those ranges because the corresponding high-rank factors also fail these properties there.

Tim de Laat and Mikael de la Salle, Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders, Journal fuer die reine und angewandte Mathematik 737 (2018), 49-69, DOI 10.1515/crelle-2015-0043, prove increasingly strong failure of the completely bounded approximation property as the rank grows. In the formulation later compiled for $\mathrm{SL}_n$, failure holds for $p>2+2/\lfloor (n-1)/2\rfloor$. Thus every $n>4$ is covered for $p>3$, whereas the established threshold for $\mathrm{SL}_3$ is only $p>4$. A positive approximation result for $L_p(\mathcal L(\mathrm{SL}_3(\mathbb Z)))$ at some $3<p\leq4$ would therefore settle the stated nonisomorphisms, but none is known.

Mikael de la Salle, Analysis on simple Lie groups and lattices, Proceedings of the International Congress of Mathematicians 2022, Volume 4 (EMS Press, 2023), 3166-3188, DOI 10.4171/ICM2022/160, explicitly presents this missing near-$2$ $L_p$-multiplier regime as the obstacle to distinguishing $\mathcal L(\mathrm{SL}_3(\mathbb Z))$ from $\mathcal L(\mathrm{PSL}_n(\mathbb Z))$ for $n>4$.

The current gap is confirmed and sharpened by two recent sources. Javier Parcet, The impact of Schur multipliers in harmonic analysis and operator algebras, Proceedings of the International Congress of Mathematicians 2026, Volume 4 (2026), DOI 10.1137/25M1803371, Problem 3.3, still asks whether the relevant Fourier critical index is greater than $2$. Mikael de la Salle, Kakeya conjecture and High-Rank Lattice von Neumann algebras, 2026, arXiv:2602.14623, Question 1.1 and Theorem 1.3, likewise leaves the noncommutative $L_p$ operator space approximation problem open and proves that a positive answer in rank $2d-1$ would imply a Kakeya-type conjecture in dimension $d$. These results underscore the difficulty but do not resolve the stated factor nonisomorphism.