Profinite Genera and Simple Kähler Groups

Definitions and mathematical statement

A Kähler group is a group isomorphic to $\pi_1(X,x)$ for a connected compact Kähler manifold $X$. A group $G$ is residually finite if, for every $g\in G\setminus\{1\}$, there exist a finite group $Q$ and a homomorphism $f:G\to Q$ with $f(g)\ne 1$. Its profinite completion is \[ \widehat G=\varprojlim_{\substack{N\trianglelefteq G\\[G:N]<\infty}}G/N, \] with its inverse-limit topology. Let $\mathcal K$ denote the class of Kähler groups and $\mathcal K_{\mathrm{rf}}$ its subclass of residually finite groups. For either comparison class $\mathcal C$, define \[ \operatorname{Gen}_{\mathcal C}(G) =\left\{[H]_{\cong}:H\in\mathcal C,\ \widehat H\cong_{\mathrm{top}}\widehat G\right\}, \] where $[H]_{\cong}$ is the abstract group isomorphism class of $H$, and $\cong_{\mathrm{top}}$ means isomorphism of topological groups.

The first supplied question asks whether \[ \forall G\in\mathcal K_{\mathrm{rf}},\qquad \left|\operatorname{Gen}_{\mathcal C}(G)\right|<\infty. \] The supplied record calls this the finiteness of the $\mathcal K$-profinite genus, but does not supply its definition. In particular, it does not establish whether the comparison class is $\mathcal C=\mathcal K_{\mathrm{rf}}$ or $\mathcal C=\mathcal K$. These are distinct candidate formulations, and the intended choice requires source clarification.

A group $S$ is simple if $S\ne\{1\}$ and its only normal subgroups are $\{1\}$ and $S$. The second supplied question, read literally, is \[ \exists S\in\mathcal K\quad S\text{ is simple}. \] The record does not explicitly require $S$ to be infinite. If infinite simplicity is intended, the formulation instead is \[ \exists S\in\mathcal K\quad \left(|S|=\infty\ \text{and}\ S\text{ is simple}\right). \] This additional condition is unresolved here. Residual finiteness is imposed on $G$ in the first question; the supplied wording imposes no such condition on $S$ in the second.

The background also mentions the related question \[ \exists H\in\mathcal K\quad H\text{ is not residually finite}. \] The supplied curation attributes a positive answer to Toledo. This is contextual information, not an additional unresolved target or an independently verified result in this draft.

Short English statement

The record asks whether every residually finite Kähler group has only finitely many Kähler groups with the same profinite completion, counted up to group isomorphism, and whether a simple Kähler group exists. The comparison class for the genus and any intended infinitude requirement for the simple group need clarification. Supplied dated curation attributes the related existence of a non-residually-finite Kähler group to Toledo.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006195$, exact source code catalogue entry; title: Profinite Genera and Simple Kähler Groups. The supplied record identifies the collection as Oberwolfach Reports Open Problems, with proposed year 2025 and no named proposer.
• Supplied report citation: Geometric Structures in Group Theory (2025), Oberwolfach Reports, pp. 275--338, DOI 10.4171/owr/2025/6. https://doi.org/10.4171/owr/2025/6.
• Supplied literature source, labelled "OWR 2025/6 Kahler-group report": https://ora.ox.ac.uk/objects/uuid%3A3212138f-4ec5-43d4-9af3-49789a6dce51/files/r6395w9580.
• Supplied literature source, labelled "Hughes et al., profinite rigidity of Kahler groups": https://arxiv.org/abs/2501.13761.
• Supplied literature source, labelled "Toledo, non-residually-finite Kahler groups": https://library.slmath.org/books/Book37/files/toledo.pdf. The extracted report attributes the positive answer to Toledo, reference [16], and describes the simple-group question as a variation of Serre's question about non-residually-finite Kähler groups.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified here. It labels the record partially_solved, reports that genus finiteness and existence of a simple Kähler group remain open in the report, and attributes a positive answer to the related non-residual-finiteness question to Toledo. Its background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. Those are supplied provenance claims; this draft does not independently verify them or claim to have visited any source.