Kirby Problem 2.30

Definitions and mathematical statement

Given a finite simple graph $\Gamma=(V,E)$, form the right-angled Artin group \[ A(\Gamma)=\langle v\ (v\in V)\mid vw=wv\text{ for }\{v,w\}\in E\rangle. \] For a closed orientable surface $S_g$ of genus $g\geq2$, its fundamental group has presentation \[ \pi_1(S_g)=\left\langle a_1,b_1,\ldots,a_g,b_g\ \middle| \prod_{i=1}^g[a_i,b_i]=1\right\rangle. \] Such groups are called closed hyperbolic surface groups.

Characterize the graphs $\Gamma$ for which there is some $g\geq2$ and an injective homomorphism $\pi_1(S_g)\hookrightarrow A(\Gamma)$. In addition, is there a terminating algorithm deciding this property from the finite adjacency data of $\Gamma$, or a necessary-and-sufficient graph-theoretic criterion? The genus is existentially quantified and is not part of the input. Realizing the surface relator by elements of $A(\Gamma)$ is not enough: the resulting homomorphism must have trivial kernel. Allowing nonorientable closed hyperbolic surfaces would not change the existence question, since their orientable double covers give subgroups of the displayed orientable type.

Short English statement

Which right-angled Artin groups contain a closed hyperbolic surface group, and can this be decided from the defining graph?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2778 (KP-2.30), 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 2.30. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.