Right-angled Artin groups on finite subgraphs of disk graphs

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