Right-angled Artin groups on finite subgraphs of disk graphs

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.