Ultraproduct II_1 Factors — Problem 9

Definitions and mathematical statement

For a discrete group $H$, let $L(H)=\{\lambda_H(h):h\in H\}''\subseteq B(\ell^2(H))$ be its group von Neumann algebra, where $(\lambda_H(h)\xi)(k)=\xi(h^{-1}k)$. Let $\mathbb F_n$ denote the free group of rank $n$. Here $\cong$ denotes isomorphism of von Neumann algebras.

Under the usual model-theoretic interpretation, an existential inclusion of $\mathbb F_2$ in $G$ is an injective group homomorphism $i:\mathbb F_2\hookrightarrow G$ such that, for every quantifier-free formula $\varphi(\bar x,\bar a)$ in the language of groups, with finite parameter tuple $\bar a$ from $\mathbb F_2$, \[ \mathbb F_2\models\exists\bar x\,\varphi(\bar x,\bar a) \quad\Longleftrightarrow\quad G\models\exists\bar x\,\varphi(\bar x,i(\bar a)). \] This interpretation of "existential inclusion" is not explicitly defined in the supplied record.

The literal first question is whether \[ \exists\,G\;\exists\,i:\mathbb F_2\hookrightarrow G\;\exists\,n\in\mathbb N \quad\text{such that}\quad i\text{ is an existential inclusion and }L(G)\not\cong L(\mathbb F_n). \] The convention for whether $0\in\mathbb N$ is not supplied. No countability or factor hypothesis on $G$ is explicitly stated.

The source then asks, specifically, whether there exists a "non elementary surface group" $G$ such that $L(G)\not\cong L(\mathbb F_n)$. The surface class and the meaning of "non elementary" are not defined in the supplied record. It is also not explicit whether this second question retains the existential-inclusion requirement or how its rank $n$ is quantified.

Unresolved qualification. The phrase "for some $n\in\mathbb N$" in the first question gives the existential rank quantifier displayed above. Requiring instead that $L(G)\not\cong L(\mathbb F_n)$ for every finite rank $n$ would be a different formulation. The supplied record does not justify replacing the literal existential quantifier by that universal quantifier. These issues require source clarification before a single precise intended formulation can be fixed.

Short English statement

The literal question asks for a group containing an existentially embedded free group of rank two whose group von Neumann algebra differs from that of a free group of some finite rank. A further question asks for a non elementary surface group with such a nonisomorphism. The intended rank quantifier, the surface-group class, and the hypotheses inherited by the second question require clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $801108$, source problem code catalogue entry, title "Ultraproduct $\mathrm{II}_1$ Factors --- Problem 9." Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attribution supplied by the registry: Elayavalli, proposed in 2025; source list "Elayavalli --- 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors (2025)," Problem 9. Supplied source URL: https://arxiv.org/abs/2511.20377.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; the source has not been independently accessed or verified for this draft.
• The registry labels this problem "open." This is a supplied status label, not an independently verified assessment of the literature.