OP-01057 Solution claimed

Fundamental group of the group factor of PSLₙ(ℤ)

For a II$_1$ factor $M$, $\mathcal F(M)$ is its fundamental group. Connes proved that $\mathcal F(L\Gamma)$ is countable for every icc group $\Gamma$ with property (T).

Problem. Is $\mathcal F(L(\mathrm{PSL}_n(\mathbb Z))) = \{1\}$ for every $n \ge 3$?

Source: A. Connes, who proved the fundamental group of a property (T) group factor countable (1980) and asked for these fundamental groups (Noncommutative geometry, 1994, Ch. V, App. B); for icc property (T) groups, whether it is trivial, S. Popa, Some open problems in W*-rigidity (2013); listed as R.3 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: fundamental group of II₁ factors, lattices in Lie groups, Peterson's operator algebra problems, property (T)

References

@article{connes1980,
  author  = {Connes, Alain},
  title   = {A factor of type {II}$_1$ with countable fundamental group},
  journal = {J. Operator Theory},
  volume  = {4},
  number  = {1},
  pages   = {151--153},
  year    = {1980},
}
@misc{peterson-r3,
  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.3; version of 15 May 2026},
}
@misc{popa2013problems-r3,
  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{chifan2024,
  author  = {Chifan, Ionu{\c t} and Das, Sayan and Houdayer, Cyril and Khan, Krishnendu},
  title   = {Examples of property {(T)} {II}$_1$ factors with trivial fundamental group},
  journal = {Amer. J. Math.},
  volume  = {146},
  number  = {2},
  pages   = {435--465},
  year    = {2024},
  doi     = {10.1353/ajm.2024.a923239},
}

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-01057, "Fundamental group of the group factor of PSLₙ(ℤ)", clopen.solutions.