Kirby Problem 3.26

Definitions and mathematical statement

For a finite graph $\Lambda$, its right-angled Artin group is $A_\Lambda=\langle V(\Lambda)\mid [u,v]=1\text{ for }\{u,v\}\in E(\Lambda)\rangle$. Let $M$ be a complete finite-volume hyperbolic three-manifold, and define \[ d(M)=\min\{[\pi_1(M):H]:H\text{ embeds in }A_\Lambda \text{ for some finite }\Lambda\}. \] This minimum is finite by virtual specialness. Choose a connected cover $M^1\to M$ of degree $d(M)$. Among the finite graphs for which $\pi_1(M^1)\hookrightarrow A_\Lambda$, measure how small $\Lambda$ can be.

Obtain effective quantitative control of this minimum complexity from explicit data for $M$, for example a triangulation of a compact core. The source proposes several separate complexity measures: number of vertices, cohomological dimension of $A_\Lambda$ (the maximum clique size), chromatic number, or graph diameter. A diameter convention must be specified for disconnected graphs; the vertex-count version has no such ambiguity.

For example, bound the least possible $|V(\Lambda)|$ for a minimal-degree cover in terms of its input triangulation. If several minimal-degree covers exist, one can seek bounds valid for each or optimize over them, explicitly stating the choice. Estimating $d(M)$ alone does not bound the Artin-group complexity asked for here.

Short English statement

Quantitatively bound the size or other graph complexity of an Artin group containing the fundamental group of a smallest suitable cover of a hyperbolic three-manifold.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2824 (KP-3.26), 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 3.26. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.