Right-angled Artin groups on finite subgraphs of disk graphs

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