Problem 2.2 — Compute CommModg (Γ) for various subgroups Γ< Modg.

Definitions and mathematical statement

Let $S_g$ be a surface of genus $g$ in the source's mapping-class convention and put $H=\operatorname{Mod}_g$. For $\Gamma<H$, its ambient commensurator is \[ \operatorname{Comm}_H(\Gamma)= \{h\in H:[\Gamma:\Gamma\cap h\Gamma h^{-1}]<\infty, [h\Gamma h^{-1}:\Gamma\cap h\Gamma h^{-1}]<\infty\}. \] This differs from the abstract commensurator of $\Gamma$. A subgroup is self-commensurating when $\operatorname{Comm}_H(\Gamma)=\Gamma$.

Problem 2.2 asks to compute $\operatorname{Comm}_{\operatorname{Mod}_g}(\Gamma)$ for various subgroups $\Gamma<\operatorname{Mod}_g$. No exhaustive class of subgroups is prescribed. The source reports that Paris--Rolfsen and Paris prove self-commensuration for most curve-stabilizing and subsurface mapping-class subgroups, but the exact exceptions and embedding conventions are not included. It motivates self-commensuration by the irreducibility of representations induced to $H$ from finite-dimensional irreducible unitary representations of $\Gamma$, giving an injection from that finite-dimensional unitary dual into the unitary dual of $H$.

The extracted record also contains the following subsequent background on volumes of representations, rather than a further numbered requirement of Problem 2.2. For a fixed finitely generated group $G$, write $X_g(G)=\operatorname{Hom}(G,\operatorname{Mod}_g)/\operatorname{Mod}_g$, quotienting by conjugation. If $G=\pi_1(X)$ for a smooth closed oriented $n$-manifold and $\mathcal M_g$ is interpreted as an appropriate orbifold classifying space for $\operatorname{Mod}_g$, a representation determines a homotopy class $f:X\to\mathcal M_g$. For $[\xi]\in H^n(\mathcal M_g;{\mathbb{R}})$ the source's invariant is \[ \nu_\xi([\rho])=\langle f^*[\xi],[X]\rangle, \] written there as $\int_X f^*\xi$. For $n=2k$ it defines \[ \operatorname{Vol}_k([\rho])=\int_X f^*(\omega_{\mathrm{WP}}^k), \] using the Weil--Petersson symplectic form, with no factorial normalization. Integration requires an orientation, absent from the literal excerpt. The ordinary coarse moduli space is not automatically a classifying space; the orbifold interpretation and relevant genus conditions must be supplied.

Short English statement

Compute ambient commensurators of specified mapping-class subgroups, particularly which are self-commensurating. The extracted following discussion of representation volumes is retained as contextual material with its missing moduli and orientation conventions flagged.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000009$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, edited volume, Problem 2.2, PDF page 23: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; 329 labeled headings with page-disambiguated numbering.
• The excerpt attributes commensurator results to Paris--Rolfsen and Paris and cites label [Pa], whose bibliography entry is not supplied. The registry mentions work for Torelli, Veech and surface subgroups without precise references.