Problem 3.1 — (a) Is it possible to generate the mapping class group of a closed nonori- entable surface by two elements?

Definitions and mathematical statement

Let $N_h=\#^h{\mathbb{R}} P^2$ be a closed connected nonorientable surface of genus $h$, and let $\operatorname{Mod}(N_h)=\pi_0\operatorname{Diff}(N_h)$ be its full mapping class group. No orientation-preserving subgroup is selected on this nonorientable surface. For the genera intended in the source, the compound problem asks three generation questions:

1. Do there exist $a,b\in\operatorname{Mod}(N_h)$ with $\langle a,b\rangle=\operatorname{Mod}(N_h)$?
2. Can $a,b$ be chosen to have finite order?
3. Do there exist involutions $u,v,w$, each of order two, with $\langle u,v,w\rangle=\operatorname{Mod}(N_h)$?

The second is a two-torsion-generator question, not merely a finite generating set consisting of torsion elements. The third fixes the number three and requires order-two elements; it does not restrict them to any particular geometric kind of involution. These conditions are on generators of the entire mapping class group, not a finite-index subgroup.

The source does not state a genus threshold or exceptions, nor whether "by two/three" means at most that many generators or a set of that exact size. The displayed existence statements use the customary bounded-generator reading without demanding that the group have minimal rank exactly two or three. Small-genus cases and whether identity is allowed in a torsion tuple need the intended convention; genuine involutions above have order exactly two. Thus no single all-$h\geq1$ affirmative assertion is silently appended.

Short English statement

Can a closed nonorientable surface's full mapping class group be generated by two elements, by two finite-order elements, or by three involutions? The genus range and small-case conventions are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID $11000119$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 3.1, PDF page 98, parts (a)--(b): https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• The research summary refers generally to later generation work without any precise citation.