Right-angled Artin groups on finite subgraphs of disk graphs

Abstract

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