Conjecture 4.2 — ( Mg is maximal).

Definitions and mathematical statement

For $g\geq3$, let $\mathcal M_g$ be the moduli orbifold of closed genus-$g$ Riemann surfaces, modeled by $\operatorname{Teich}_g/\operatorname{Mod}_g$ with its stabilizer data. An orbifold covering respects local uniformizing charts and isotropy; a finite covering has finitely many sheets.

The printed maximality conjecture says that $\mathcal M_g$ does not finitely orbifold-cover any other smooth orbifold. In the usual no-proper-quotient interpretation, this asks that every connected finite orbifold quotient $p:\mathcal M_g\to\mathcal O$ be trivial. Whether other means merely a nonisomorphic target, or excludes every covering of degree greater than $1$, needs clarification.

The source also proposes a stronger statement: if $N\to\mathcal M_g$ is any finite orbifold cover and $N\to\mathcal O$ is an orbifold cover, then $\mathcal O$ should itself orbifold-cover $\mathcal M_g$. The second covering is not explicitly stated finite, and the text does not require that these maps form a compatible factorization of the given map $N\to\mathcal M_g$. This stronger statement is explicitly presented only as something that may be true.

The smooth-category formulation has a material unresolved issue. Complex conjugation gives a natural involution on the moduli orbifold, or equivalently one may enlarge to the extended mapping class group; allowing that quotient produces a natural smooth orbifold quotient of degree $2$. Thus the intended allowed category must explain whether such quotients are excluded, for example by additional complex/holomorphic restrictions. No such restriction is printed, and it is not silently inserted. The following related topology question is mentioned but not included.

Short English statement

Is the moduli orbifold maximal among the allowed finite orbifold quotients for $g\geq3$? The proposed stronger cover statement and the missing restriction excluding conjugation quotients require clarification.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Conjecture 4.2, "$\mathcal M_g$ is maximal," supplied PDF page 38. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. No separate reference is supplied for this item.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000041$, exact source code catalogue entry; supplied OPEN-TRIAGE summary says status cannot be verified without the statement.