OP-01055 Solution claimed

Connes's rigidity conjecture for lattices in higher-rank simple Lie groups

Problem. Let $G$ be a simple Lie group of real rank at least $2$ with trivial centre, and $\Gamma < G$ a lattice. If $\Lambda$ is a group with $L\Gamma \cong L\Lambda$, must $\Gamma$ and $\Lambda$ be isomorphic?

Source: A. Connes, Classification des facteurs (1982), whose rigidity conjecture this is, for lattices; listed as R.1 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: Connes's rigidity conjecture, lattices in Lie groups, Peterson's operator algebra problems, W*-rigidity

References

@incollection{connes1982-r1,
  author    = {Connes, Alain},
  title     = {Classification des facteurs},
  booktitle = {Operator algebras and applications, Part 2 (Kingston, Ont., 1980)},
  series    = {Proc. Sympos. Pure Math.},
  volume    = {38},
  publisher = {Amer. Math. Soc.},
  pages     = {43--109},
  year      = {1982},
  doi       = {10.1090/pspum/038.2/679497},
}
@misc{peterson-r1,
  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.1; version of 15 May 2026},
}
@inproceedings{houdayer2023,
  author    = {Houdayer, Cyril},
  title     = {Noncommutative ergodic theory of higher rank lattices},
  booktitle = {Proceedings of the International Congress of Mathematicians 2022, Vol. 4},
  publisher = {EMS Press},
  pages     = {3202--3223},
  year      = {2023},
  doi       = {10.4171/ICM2022/39},
}
@article{houdayerioana2026,
  author  = {Houdayer, Cyril and Ioana, Adrian},
  title   = {Weyl groups and rigidity of von {N}eumann algebras},
  journal = {Tunis. J. Math.},
  volume  = {8},
  number  = {4},
  pages   = {691--704},
  year    = {2026},
  doi     = {10.2140/tunis.2026.8.691},
}
@article{cowling1989,
  author  = {Cowling, Michael and Haagerup, Uffe},
  title   = {Completely bounded multipliers of the {F}ourier algebra of a simple {L}ie group of real rank one},
  journal = {Invent. Math.},
  volume  = {96},
  number  = {3},
  pages   = {507--549},
  year    = {1989},
  doi     = {10.1007/BF01393695},
}

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-01055, "Connes's rigidity conjecture for lattices in higher-rank simple Lie groups", clopen.solutions.