de la Harpe–Voiculescu conjecture on Fuchsian-group factors

For every finitely generated, torsion-free, non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$ , the group von Neumann algebra $L(\Gamma)$ is isomorphic to a free group factor; in particular, for the fundamental group of a closed orientable surface of genus $g\geq 2$ , $L(\pi_{1}(\Sigma_{g}))\cong L(\mathbb{F}_{2g-1})$ .

References

Primary source

arXiv

Additional references

On the II₁ Factors of Fuchsian Groups — arXiv — D. Shlyakhtenko

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture, but the result has not been independently verified.

The de la Harpe–Voiculescu conjecture concerns whether the relevant Fuchsian-group factors have the predicted free-group-factor classification. D. Shlyakhtenko’s preprint claims this classification follows from a free-complementation result for a commutator in $\mathrm{L}(\mathbb{F}_2)$ .

September 2026 claimed solution

On September 10, 2026, D. Shlyakhtenko’s preprint On the II₁ Factors of Fuchsian Groups reported the claimed resolution. It explicitly credits OpenAI’s ChatGPT Pro 6.0 with obtaining the result, but the mathematical claim remains unrefereed and unverified.

Current status (as of September 2026): a preprint claims the conjecture is solved, but independent mathematical verification is not recorded.

OpenAI's ChatGPT Pro 6.0solved2026-09-10evidence
De la Harpe–Voiculescu free-group-factor conjecture claimed solved

Sources

No solutions have been posted yet.