Problem 2.4 — Let Γ be a subgroup of finite index in the mapping class group Mod1,2 and let φ: Γ → Γ be an automorphism.

Definitions and mathematical statement

Let $G=\operatorname{Mod}_{1,2}$. In the customary finite-type surface convention this is the mapping class group of a genus-one surface with two punctures or marked points; the excerpt does not specify whether they may be permuted, whether orientation reversal is allowed, or whether a boundary convention is intended instead.

Problem 2.4. For every subgroup $\Gamma\le G$ with $[G:\Gamma]<\infty$ and every group automorphism $\varphi:\Gamma\to\Gamma$, does there exist $\Phi\in\operatorname{Aut}(G)$ such that \[ \Phi|_{\Gamma}=\varphi? \] In particular, an extension must preserve $\Gamma$ setwise. Equivalently, is the restriction map \[ \{\Phi\in\operatorname{Aut}(G):\Phi(\Gamma)=\Gamma\} \longrightarrow\operatorname{Aut}(\Gamma) \] surjective for every finite-index $\Gamma$? No normality or torsion-freeness hypothesis is imposed on $\Gamma$. This is an extension to an abstract automorphism of the full group, not an assertion that every automorphism of $\Gamma$ must be inner in $\Gamma$.

As subsequent context, the source reports a theorem of Harvey and the chapter author [14]: for $g>h$, a homomorphism $\operatorname{Mod}_g\to\operatorname{Mod}_h$ has trivial image except that, for $g=2$, its image is allowed to have order at most two. That reported result concerns homomorphisms between different genus groups; it is not the finite-index extension assertion posed here. The source then points to a following question, whose text is not included in this record.

Short English statement

Does every automorphism of every finite-index subgroup of $\operatorname{Mod}_{1,2}$ extend to an automorphism of the whole group? The exact puncture and orientation conventions for this exceptional surface group need confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 11000109, source code catalogue entry; no individual proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Problem 2.4, PDF page 94; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; its inventory distinguishes 329 labeled headings by source page.
• Harvey and the chapter author, source reference [14], for the subsequently reported different-genus homomorphism theorem. The full citation and exact mapping-class conventions are not supplied.