Questions in Geometric Group Theory — Q 12.22

Definitions and mathematical statement

Let $S_{g,b,n}$ be a compact connected orientable surface of genus $g\geq0$ with $b\geq0$ boundary circles and $n\geq0$ marked punctures in its interior. Its mapping class group consists of orientation-preserving homeomorphisms fixing the boundary pointwise and preserving the puncture set, modulo isotopies with the same restrictions. Punctures may be permuted; requiring them individually fixed gives a finite-index pure subgroup.

A group $G$ is linear if there exist a field $k$, an integer $d\geq1$, and an injective homomorphism $G\to\operatorname{GL}_d(k)$. The field and matrix size are part of the existence assertion, rather than prescribed in advance.

Is $\operatorname{Mod}(S_{g,b,n})$ linear for every finite triple $(g,b,n)$? Determine the remaining cases of this question without assuming the genus is zero or restricting to special symmetry subgroups. One can distinguish the version over characteristic zero from versions over positive characteristic, but the source's unrestricted linearity question allows any field.

A homology representation need not be faithful, so the natural action on the first homology of the surface does not by itself settle linearity. Finite generation or faithful actions on infinite-dimensional vector spaces likewise do not imply the required finite-dimensional representation. The dimension of a prospective faithful representation may grow with the topological complexity of the surface. Boundary-fixing and puncture-permuting conventions are stated explicitly so that the abstract group being represented is unambiguous.

Short English statement

Do mapping class groups of all finite-type orientable surfaces admit faithful finite-dimensional linear representations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101222 (AMR-010-1222), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 12.22 (PDF page 23). Original source indexed by AMR-010-1222. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.