Problem 3.1 — Study, systematically and with the help of computers, the finite quotients of Mg which do not factor through Sp (2g, Z).

Definitions and mathematical statement

Here the source's $M_g$ denotes a mapping class group, not the moduli space. Write it as $G_g$, with homology representation \[\pi:G_g\to\operatorname{Sp}(2g,{\mathbb{Z}}),\qquad\mathcal I_g=\ker\pi.\] The programme is to study finite quotients $\phi:G_g\twoheadrightarrow Q$ that do not factor through $\pi$. When $\pi$ is the standard surjective homology representation, this means there is no $\overline\phi$ with $\phi=\overline\phi\circ\pi$, equivalently \[\mathcal I_g\not\subseteq\ker\phi.\] The task is a systematic computer-assisted investigation and organization of such quotients, not merely proof that one exists.

The source proposes starting perhaps at $g=3$ and enumerating homomorphisms $G_g\to\Sigma_n$ for increasing $n$. Given a finite presentation, assign permutations to every generator and test all defining relations. Each successful assignment defines a homomorphism and its image is a finite quotient; surjectivity onto all of $\Sigma_n$ is not required. A catalogue must specify its duplication/equivalence convention, for example conjugacy of permutation representations or isomorphism of image quotients, neither fixed in the excerpt.

The source points to [52] for a simple presentation of $M_{g,0,0}$. It reports that factorization through the symplectic quotient is tested using two additional relation types: the genus-one separating-curve twist for $g\geq2$, and the genus-one bounding-pair relation for $g\geq3$, citing [23]. In conventional terms the second is a bounding-pair map $t_a t_b^{-1}$ for homologous disjoint curves bounding the appropriate subsurface; the extracted phrase "Dehn twist on a bounding pair" needs that convention. The exact normal-generation statement and test words must be verified before treating these as a sufficient computational test.

Further necessary restrictions are reported: generators chosen as twists in nonseparating curves must have conjugate images, and [7] gives generation by six involutions, with its genus hypotheses not reproduced. These restrictions can reduce enumeration but do not alone ensure that a tuple defines a homomorphism. The claim that modern computers put the programme within reach is historical motivation, not a supplied bound or completed catalogue. The ending announces a related nonfinite-quotient problem that is absent.

The research summary reports existence of nonsymplectic finite quotients but only an incomplete "$2^{?}$" genus-three example and no usable reference. Such existence, even if verified, would not settle the systematic study requested here.

Short English statement

Systematically enumerate and understand finite mapping-class quotients that detect Torelli elements, using permutation representations and relation tests. The source asks for an organized investigation, so an isolated existence result does not finish the programme.

Sources

• Local UnsolvedMath registry, permanent ID $11000172$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 3.1, PDF page 160: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Internal references [52] (presentation), [23] (symplectic-factorization relations) and [7] (six involutions) lack full entries. The summary's unspecified genus-three quotient has no exact order or primary citation.