Kirby Problem 2.27

Definitions and mathematical statement

The braid group $B_n$ is generated by $\sigma_1,\ldots,\sigma_{n-1}$, with relations $\sigma_i\sigma_j=\sigma_j\sigma_i$ for $|i-j|\geq2$ and $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$. For a finite simple graph $\Gamma$ with vertex set $V$ and edge set $E$, its right-angled Artin group is \[ A(\Gamma)=\langle v\ (v\in V)\mid [v,w]=1\text{ when }\{v,w\}\in E\rangle, \qquad [v,w]=vwv^{-1}w^{-1}. \] There are no further defining relations.

For every integer $n\geq4$, does there exist a subgroup $H\leq B_n$ of finite index, a finite graph $\Gamma$, and an injective homomorphism $H\hookrightarrow A(\Gamma)$? Finite index means that $B_n$ has finitely many cosets of $H$. Both $H$ and $\Gamma$ may depend on $n$; no uniform graph or bound on the index is demanded. The subgroup need not equal the full braid group. Conversely, finding a RAAG-embeddable subgroup of infinite index would not establish the required virtual embedding. Injectivity must hold for all elements of the chosen finite-index subgroup.

Short English statement

Does each braid group on at least four strands have a finite-index subgroup that embeds in a right-angled Artin group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2775 (KP-2.27), 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 2.27. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.