Question 6.2 — (Rigidity of the Mod g,1 action on S1).

Definitions and mathematical statement

Let $\operatorname{Mod}_{g,1}$ denote the source's mapping class group and let $\rho_{\mathrm{std}}:\operatorname{Mod}_{g,1}\to\operatorname{Homeo}^+(S^1)$ be the standard action referenced in its missing equation (19). The assertion that this group has torsion indicates a marked-point or puncture convention rather than a boundary-fixed convention, but the exact definition and genus range must be recovered.

Question 6.2. Is every faithful action $\rho:\operatorname{Mod}_{g,1}\to\operatorname{Homeo}^+(S^1)$ conjugate to $\rho_{\mathrm{std}}$ by an orientation-preserving circle homeomorphism? Explicitly, must there be $h\in\operatorname{Homeo}^+(S^1)$ with \[ \rho(\gamma)=h\rho_{\mathrm{std}}(\gamma)h^{-1} \quad\text{for every }\gamma\in\operatorname{Mod}_{g,1}? \] Ask the same question for every finite-index subgroup $\Gamma\leq\operatorname{Mod}_{g,1}$, comparing a faithful action of $\Gamma$ with $\rho_{\mathrm{std}}|_\Gamma$. The source asks exact conjugacy in $\operatorname{Homeo}^+(S^1)$, not semiconjugacy or conjugacy allowing reversal of the circle orientation, and does not impose minimality. These distinctions are essential and are not silently weakened.

The source proposes a stronger dynamical characterization analogous to the Convergence Groups Theorem of Tukia, Casson--Jungreis, and Gabai, replacing asymptotic source--sink behavior by alternating source--sink--$\cdots$--source--sink behavior. No precise sequential condition, number of alternations, or refinement is specified, so this remains a suggested research direction.

Its context describes the lift group \[ \widetilde{\operatorname{Homeo}}^+(S^1) =\{f\in\operatorname{Homeo}^+({\mathbb{R}}):f(x+1)=f(x)+1\}, \] with the central extension by integer translations. Pullback through $\rho_{\mathrm{std}}$ gives \[ 1\longrightarrow{\mathbb{Z}}\longrightarrow \widetilde{\operatorname{Mod}}_{g,1} \longrightarrow\operatorname{Mod}_{g,1}\longrightarrow1. \tag{20} \] The source uses the torsion-free lift group and torsion in $\operatorname{Mod}_{g,1}$ to say (20) does not split, giving a nonzero class $\xi\in H^2(\operatorname{Mod}_{g,1};{\mathbb{Z}})$, identified with the Euler cocycle. It describes evaluation using pointed maps $\Sigma_h\to\mathcal M_g$ and a pulled-back "universal circle bundle." The relevant orbifold, bundle, and pointed/unpointed base conventions are not supplied, so that geometric interpretation requires review.

Finally, it notes that torsion obstructs left-orderings and faithful orientation-preserving actions on ${\mathbb{R}}$ for these mapping class groups, says this is the only obstruction then known to the author, and notes that torsion disappears in appropriate finite-index subgroups. The last statement does not assert that the resulting torsion-free subgroups are left-orderable or have faithful line actions.

Short English statement

Ask whether every faithful circle action of the pointed mapping class group, and of its finite-index subgroups, is conjugate by an orientation-preserving homeomorphism to the specified standard action. Exact conjugacy, semiconjugacy, and minimal-action variants must be distinguished; the source also records the lift extension and Euler-class context.

Sources

• Local UnsolvedMath registry, permanent ID 11000071, source code catalogue entry, Question 6.2, "Rigidity of the $\operatorname{Mod}_{g,1}$ action on $S^1$," PDF page 54. No individual proposer is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across chapters, with repeated numbers disambiguated by page. The defining equation (19) is absent; the central extension is labeled (20).
• The statement credits Tukia, Casson--Jungreis, and Gabai for the convergence-groups theorem. The summary mentions Matsumoto and unspecified circle-action results, with an unresolved genus placeholder, but gives no complete theorem or bibliography.
• Supplied status open and OPEN-TRIAGE. Its summary says the specific action-rigidity formulation was not fully verified. Historical phrases such as "as far as we know" about line actions are not treated as a 2026 audit. Current primary-literature status and redistribution terms remain NEEDS_REVIEW.