Kirby Problem 2.28

Definitions and mathematical statement

For a finite simplicial graph $\Gamma$, the right-angled Artin group is \[ A_\Gamma=\langle v\in V(\Gamma)\mid uv=vu\text{ whenever }\{u,v\}\in E(\Gamma)\rangle. \] A nontrivial join is a graph $J=J_1*J_2$ with both vertex sets nonempty, containing every edge between $J_1$ and $J_2$; its Artin group is the direct product $A_{J_1}\times A_{J_2}$. A join subgroup means the subgroup generated by the vertices of an induced subgraph of this form. Assume $\Gamma$ itself is not a nontrivial join.

Call a nonidentity element loxodromic in the RAAG sense used here if it is not conjugate into any join subgroup. Equivalently its cyclically reduced support is not contained in an induced nontrivial join. This is the algebraic criterion for the usual rank-one elements, avoiding a choice of conjugate word.

Is there an oriented finite-type surface $S$ and an injective homomorphism $\iota:A_\Gamma\to\operatorname{Mod}(S)$ that sends every such element to a pseudo-Anosov mapping class? A pseudo-Anosov preserves transverse measured singular foliations with reciprocal stretching factors $\lambda,\lambda^{-1}$, $\lambda>1$. The surface and embedding may depend on $\Gamma$, but one embedding must satisfy the condition for all loxodromic elements simultaneously.

Short English statement

Embed each right-angled Artin group that does not split as a direct product into a mapping class group so that every rank-one element becomes pseudo-Anosov.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2776 (KP-2.28), 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 2.28. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.