Problem 4.4 — Does there exist a simple cycle of dihedral subgroups of MCG (S) so that the associated reflection group P injects in…

Definitions and mathematical statement

A dihedral group is generated by two involutions; in the finite case their product has finite order $m$ and the standard presentation is $\langle r,s\mid r^2=s^2=(rs)^m=1\rangle$. The source does not specify which finite or infinite dihedral groups its construction permits.

Problem 4.4 asks whether there exists a simple cycle of dihedral subgroups of $\operatorname{MCG}(S)$ for which the associated reflection group $P$ injects into $\operatorname{MCG}(S)$. Writing $\iota:P\to\operatorname{MCG}(S)$ for the homomorphism arising from that construction, the first requested property is \[ \ker\iota=\{1\}. \] It additionally asks for such a construction whose image answers any of the questions in Problem 4.3.

The supplied record does not define simple cycle, the subgroup intersection/gluing data, the associated reflection-group presentation, or $\iota$. No particular Coxeter polygon or extra relators can be substituted from the word reflection alone. Problem 4.3 is absent, so the additional requested property of $\iota(P)$ is unidentified; it is not replaced by a guessed geometric or algebraic subgroup property. The surface type and genus range are also not supplied.

Short English statement

Can a simple cycle of dihedral mapping-class subgroups produce an injective associated reflection group, and can its image meet one of the omitted targets in Problem 4.3? The construction and targets are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000238$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 4.4, PDF page 274: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• The preceding cycle/reflection construction and Problem 4.3 are not included. No additional authoritative citation or individual attribution is supplied; the registry cannot verify the current status.