Question 2.8 — (Normal subgroups).

Definitions and mathematical statement

Let $S_g$ be a closed orientable genus-$g$ surface with $g\ge3$, let $\operatorname{Mod}_g$ be its orientation-preserving mapping class group, and let \[ \mathcal I_g=\ker\left(\operatorname{Mod}_g\longrightarrow \operatorname{Aut}(H_1(S_g;{\mathbb{Z}}))\right) \] be the Torelli group. Two subgroups $A,B$ of a common group are commensurable within that group if $A\cap B$ has finite index in both. Abstract commensurability instead asks only for isomorphic finite-index subgroups; the source does not explicitly choose between these meanings.

Question 2.8 (normal subgroups). If $\Gamma\triangleleft\operatorname{Mod}_g$ is finitely generated, must it be commensurable with $\operatorname{Mod}_g$ or with $\mathcal I_g$? Under the within-group convention, the proposed dichotomy for the intended nontrivial groups is \[ [\operatorname{Mod}_g:\Gamma]<\infty \quad\text{or}\quad [\Gamma:\Gamma\cap\mathcal I_g]<\infty\ \text{ and }\ [\mathcal I_g:\Gamma\cap\mathcal I_g]<\infty. \] A necessary qualification is missing from the literal excerpt: the trivial group is finitely generated and normal but is not commensurable with either of these infinite groups. Thus the substantive question needs a nontriviality or appropriate infinitude restriction; this draft records that issue rather than silently adding it to the source sentence.

The accompanying discussion considers \[ D_n=\langle T_c^n:c\text{ an essential simple closed curve on }S_g\rangle \] and the normal closure of a single mapping class, \[ N_\varphi=\langle h\varphi h^{-1}:h\in\operatorname{Mod}_g\rangle. \] It describes these as sources of infinitely generated normal subgroups and asks how to determine the algebraic structure of $N_\varphi$, especially whether it has finite index in $\operatorname{Mod}_g$ or in one of the groups written $\mathcal I_g(k)$. The excerpt supplies neither the restrictions on $n$ or $\varphi$ needed for its infinite-generation description nor the definition of $\mathcal I_g(k)$. In particular, no universal infinite-generation assertion for all powers or all normal closures is adopted. Its final reference to a "following" basic test question is a forward reference, not an included test problem.

Short English statement

Apart from the omitted trivial-group exception, must every finitely generated normal subgroup of a genus-at-least-three mapping class group be commensurable with the whole group or the Torelli group? The source also highlights the structure and finite-index behavior of single-element normal closures.

Sources

• Local UnsolvedMath registry, permanent ID 11000013, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Question 2.8, PDF page 27; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29. The volume inventory has 329 explicitly labeled Problem, Conjecture, and Question headings, with repeated numbering distinguished by PDF page.
• The research summary reports advances on normal subgroups since 2006 but provides no individual primary references establishing the status of this finitely generated dichotomy.