Question — Are there any other relations between N -th powers Tγ = tN γ of Dehn twists for sufficiently high N?

Definitions and mathematical statement

For the source's surface $S$, let $\mathcal C(S)$ be the allowed isotopy classes of curves, let $t_\gamma$ be the corresponding Dehn twists, and set \[ T_\gamma=t_\gamma^N,\qquad G_N=\langle t_\gamma^N:\gamma\in\mathcal C(S)\rangle \leq\operatorname{Mod}(S). \] This is the subgroup generated by the powers, not the quotient obtained by imposing $t_\gamma^N=1$.

The question on PDF page 87 asks whether, for sufficiently high positive $N$, the previously displayed relations give a presentation of $G_N$, and if not, what additional relations are required. Those "above relations" are not included in the supplied record.

Precisely, let $F$ be the free group on symbols $x_\gamma$ and let $q_N:F\twoheadrightarrow G_N$ send $x_\gamma$ to $t_\gamma^N$. If $R_N$ is the normal closure of the omitted supplied relators, the question is whether \[ \ker q_N=R_N \] for sufficiently large $N$, and otherwise to describe further relators normally generating the missing kernel. The relator family and hence $R_N$ remain placeholders tied to the source, not invented commutation or braid relations. No finite generating set or finite presentation is requested by the excerpt. Surface type, permitted curve types, and the threshold and its parameter dependence are also unstated.

Short English statement

Do the omitted proposed relations account for all relations among sufficiently high powers of Dehn twists? If they do not, determine the additional relations for the subgroup those powers generate.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000104$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, unnumbered Question, PDF page 87: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; heading disambiguated by page among 329 headings.
• The preceding relations are absent. The registry mentions congruence/small-cancellation and Korkmaz-related normal-closure work without precise papers and explicitly cannot identify a decisive matching result.