Embeddability of right-angled Artin groups on complements of trees

摘要

For a finite simplicial graph Γ, let A(Γ) denote the right-angled Artin group on Γ. Recently, Kim and Koberda introduced the extension graph Γe for Γ, and established the Extension Graph Theorem: for finite simplicial graphs Γ1 and Γ2, if Γ1 embeds into Γ2e as an induced subgraph then A(Γ1) embeds into A(Γ2). In this paper, we show that the converse of this theorem does not hold for the case Γ1 is the complement of a tree and for the case Γ2 is the complement of a path graph.