Problem 3.4 — Find a candidate for a faithful finite-dimensional matrix representation of Mg orMg,1,0.

Definitions and mathematical statement

Let $M_g$ and $M_{g,1,0}$ be the mapping class groups denoted by those symbols in the source. The excerpt does not restate their boundary/marked-point conventions, in particular the ordering of the extra indices.

Problem 3.4 asks for a candidate faithful finite-dimensional matrix representation of either group. For $G=M_g$ or $M_{g,1,0}$ this means an explicitly described homomorphism \[ \rho:G\longrightarrow\operatorname{GL}_d(k),\qquad d<\infty, \] with a specified coefficient field $k$, proposed with the goal $\ker\rho=\{1\}$. The source asks for a candidate, not merely an unsupported claim of faithfulness, and does not insist that injectivity already be proved as part of proposing it. A proposal must still define an actual representation, with its matrices or construction respecting the group's relations.

The field, characteristic, genus range and whether dimension is allowed to depend on genus are not specified. No uniform bound in $g$ is requested. The registry reports faithful representations in genus $2$, attributed to Bigelow--Budney and Korkmaz, and reports linearity for $g\geq3$ still open, with a cross-reference to source code AMR-109-0206. The scope of those reports for the differently marked/bounded group $M_{g,1,0}$ requires separate confirmation.

Short English statement

Propose an explicit finite-dimensional matrix representation of the indicated mapping class group that could be faithful. The registry reports genus-two examples, while the higher-genus and marked-surface scope needs review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000175$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 3.4, PDF page 160: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• Supplied progress attributions: Bigelow--Budney and Korkmaz for genus $2$, with a pointer to source code AMR-109-0206 for citations. Exact publications and the conventions for $M_{g,1,0}$ are not included in this record.