OP-01056 Solution claimed

Popa's strong form of Connes's rigidity conjecture for higher-rank lattices

Here $(u_g)_{g \in \Gamma}$ and $(v_h)_{h \in \Lambda}$ are the canonical unitaries of $L\Gamma$ and $L\Lambda$, and $(L\Lambda)^t$ is the amplification of $L\Lambda$ by $t$.

Problem. Let $G$ be a simple Lie group of real rank at least $2$ with trivial centre, $\Gamma < G$ a lattice, $\Lambda$ a group, and $\theta : L\Gamma \to (L\Lambda)^t$ an isomorphism for some $t > 0$. Must $t = 1$, and must there be a group isomorphism $\alpha : \Gamma \to \Lambda$, a character $\chi : \Gamma \to \mathbb T$ and a unitary $u \in \mathcal U(L\Gamma)$ such that $\theta(u u_g u^*) = \chi(g) v_{\alpha(g)}$ for every $g \in \Gamma$?

Source: S. Popa, Deformation and rigidity for group actions and von Neumann algebras (ICM 2006), (3.2′), after V. F. R. Jones, Ten problems (2000); listed as R.2 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Secondary: GR
Keywords: amplifications, Connes's rigidity conjecture, lattices in Lie groups, Peterson's operator algebra problems, W*-superrigidity

References

@inproceedings{popa2007icm,
  author    = {Popa, Sorin},
  title     = {Deformation and rigidity for group actions and von {N}eumann algebras},
  booktitle = {International Congress of Mathematicians, Vol. I (Madrid, 2006)},
  publisher = {Eur. Math. Soc.},
  pages     = {445--477},
  year      = {2007},
  doi       = {10.4171/022-1/18},
}
@incollection{jones2000,
  author    = {Jones, Vaughan F. R.},
  title     = {Ten problems},
  booktitle = {Mathematics: frontiers and perspectives},
  publisher = {Amer. Math. Soc.},
  pages     = {79--91},
  year      = {2000},
}
@misc{peterson-r2,
  author       = {Peterson, Jesse},
  title        = {Open problems in operator algebras},
  howpublished = {Web page},
  year         = {2026},
  url          = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
  note         = {Problem R.2; version of 15 May 2026},
}
@misc{popa2013problems-r2,
  author       = {Popa, Sorin},
  title        = {Some open problems in {W}$^*$-rigidity},
  howpublished = {Slides, Paris, June 2013},
  year         = {2013},
  url          = {https://www.math.ucla.edu/~popa/ProblemsJune2013.pdf},
}
@article{chifan2026out-r2,
  author  = {Chifan, Ionu{\c t} and Ioana, Adrian and Osin, Denis and Sun, Bin},
  title   = {Wreath-like products of groups and their von {N}eumann algebras {II}: outer automorphisms},
  journal = {Duke Math. J.},
  volume  = {175},
  number  = {2},
  pages   = {287--359},
  year    = {2026},
  doi     = {10.1215/00127094-2025-0028},
}

Resolution claims

The arithmetic category and stable recovery of lattice factors v1 GitHub ↗

OpenAI

We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property-$(T)$ lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners. We compute …

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: math @ 7ef5c0c, build/paper.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01056, "Popa's strong form of Connes's rigidity conjecture for higher-rank lattices", clopen.solutions.