Right-angled Artin groups on finite subgraphs of disk graphs

TL;DR

This paper explores conditions under which right-angled Artin groups associated with subgraphs of disk graphs of handlebodies embed into the handlebody groups, extending known results from curve graphs of surfaces.

Contribution

It establishes that subgraphs of disk graphs of handlebodies induce right-angled Artin groups that embed into the handlebody groups, and identifies cases where this embedding occurs despite the subgraph not being contained in the disk graph.

Findings

A subgraph of a disk graph induces an RAAG that embeds into the handlebody group.

There exist graphs not contained in any disk graph but still induce RAAGs that embed into handlebody groups.

Abstract

Koberda proved that if a graph $\Gamma$ is a full subgraph of a curve graph $\mathcal{C}(S)$ of an orientable surface $S$, then the right-angled Artin group $A(\Gamma)$ on $\Gamma$ is a subgroup of the mapping class group ${\rm Mod}(S)$ of $S$. On the other hand, for a sufficiently complicated surface $S$, Kim-Koberda gave a graph $\Gamma$ which is not contained in $\mathcal{C}(S)$, but $A(\Gamma)$ is a subgroup of ${\rm Mod}(S)$. In this paper, we prove that if $\Gamma$ is a full subgraph of a disk graph $\mathcal{D}(H)$ of a handlebody $H$, then $A(\Gamma)$ is a subgroup of the handlebody group ${\rm Mod}(H)$ of $H$. Further, we show that there is a graph $\Gamma$ which is not contained in some disk graphs, but $A(\Gamma)$ is a subgroup of the corresponding handlebody groups.