Conjecture — For every finitely generated subgroups G of ModS, the group Φf (G) is nilpotent.

Definitions and mathematical statement

Let $S$ be the source's surface and $G\le\operatorname{Mod}_S$ any finitely generated subgroup. The notation $\Phi_f(G)$ denotes an associated group in the source, but its definition is not supplied in this extract. The research summary's tentative "fixed point / finitely presented kernel" interpretation is speculation, not a definition, and is not adopted.

For any group $A$, set $\gamma_1(A)=A$ and $\gamma_{j+1}(A)=[A,\gamma_j(A)]$. It is nilpotent when $\gamma_{c+1}(A)=\{1\}$ for some finite $c$. The conjecture on PDF page 88 states \[ \forall G\le\operatorname{Mod}_S\text{ finitely generated}\quad \exists c=c(G)<\infty:\quad \gamma_{c+1}(\Phi_f(G))=\{1\}. \] The nilpotency class may depend on $G$; no uniform bound is claimed. The quantifier is over every finitely generated subgroup, not only finite-index or finitely presented subgroups. The source emphasizes limited understanding of finite-index subgroups as motivation, and calls this an old problem, referring to [I4], Section 10.10. A source-complete conjecture requires the definition of $\Phi_f$ and the surface hypotheses from that reference.

Short English statement

For every finitely generated mapping-class subgroup, the source-defined group $\Phi_f(G)$ is conjectured nilpotent, with no uniform class bound stated. Its definition is missing, so the summary's tentative interpretation is not used to invent the target.

Sources

• Local UnsolvedMath registry, permanent ID $11000107$, exact source code catalogue entry, unnumbered Conjecture, PDF page 88. No individual proposer or proposed year is supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume contains 329 explicitly typeset Problem, Conjecture and Question headings, with page-specific disambiguation used here.
• Internal reference [I4], Section 10.10, for further discussion. The bibliography entry and definition of $\Phi_f$ are absent; no author or title is guessed.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; the summary explicitly cannot verify the target and offers only a tentative interpretation of its symbol.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.