Kirby Problem 2.29

Definitions and mathematical statement

Let $I$ be a finite set and let $(m_{ij})$ be a symmetric Coxeter matrix: $m_{ii}=1$ and $m_{ij}\in\{2,3,\ldots\}\cup\{\infty\}$ for $i\ne j$. The associated Artin group is generated by $s_i$ for $i\in I$, subject to \[ \underbrace{s_i s_j s_i\cdots}_{m_{ij}\text{ letters}} =\underbrace{s_j s_i s_j\cdots}_{m_{ij}\text{ letters}} \qquad(m_{ij}<\infty). \] There is no relation for $m_{ij}=\infty$. Its Coxeter group is obtained by also imposing $s_i^2=1$. The Artin group is of finite type when this Coxeter group is finite; finite type does not mean that the Artin group itself is finite. Finite type is only a special case of the question below.

Which Artin groups on a finite generating set admit an injective homomorphism into $\operatorname{Mod}(S)$ for some oriented surface $S$ of finite type? Here mapping classes are orientation-preserving homeomorphisms modulo isotopy, with any specified boundary fixed pointwise. The surface may depend on the Artin group. The task is an existence and classification question: no restriction is imposed that the displayed generators must map to individual Dehn twists. A representation having a kernel, or an embedding of only a finite-index subgroup, does not by itself answer this question.

Short English statement

Determine which finitely generated Artin groups embed faithfully into a surface mapping class group.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2777 (KP-2.29), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.29. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.