On the II${}_1$ Factors of Fuchsian Groups

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\ge 2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=AB{A}^{-1}{B}^{-1}$ of the free group ${\mathbb{F}}_2=⟨A,B⟩$ is freely complemented in the group factor: $L({\mathbb{F}}_2)=W^{\ast }(w)\ast W^{\ast }(v)$ for some Haar unitary $v\in L({\mathbb{F}}_2)$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\mathrm{\Gamma }\subset PS{L}_2(\mathbb{R})$, $L(\mathrm{\Gamma })$ 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.