PreprintCC BY 4.0arXiv
On the II1 Factors of Fuchsian Groups
Abstract
We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus g≥2 are free group factors on 2g−1generators. The key technical ingredient involves a proof that the element w=ABA−1B−1 of the free group 𝔽2=⟨ A,B⟩ is freely complemented in the group factor: L(𝔽2)=W*(w)*W*(v) for some Haar unitary v∈ L(𝔽2) that is freely independent from w. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup Γ⊂ PSL2(ℝ), L(Γ) is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.
Related
- Quasi-Fuchsian groups and complex realisations of q-deformed real numbers
- Medvedev degrees of SFTs on cocompact Fuchsian groups
- Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups
- Limit theorems for cuspidal orbits on Fuchsian groups
- Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
BibTeX
@misc{shlyakhtenko2026factors,
title = {{On the II$_1$ Factors of Fuchsian Groups}},
author = {D. Shlyakhtenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11074},
eprint = {2609.11074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11074},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the II_1 Factors of Fuchsian Groups
AU - D. Shlyakhtenko
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11074
UR - https://arxiv.org/abs/2609.11074
AB - We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus g≥2 are free group factors on 2g−1generators. The key technical ingredient involves a proof that the element w=ABA^−1B^−1 of the free group 𝔽_2=⟨ A,B⟩ is freely complemented in the group factor: L(𝔽_2)=W^*(w)*W^*(v) for some Haar unitary v∈ L(𝔽_2) that is freely independent from w. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup Γ⊂ PSL_2(ℝ), L(Γ) is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.
ER -