Embeddability of right-angled Artin groups on the complements of linear forests

Abstract

Abstract. In this article, we prove that embeddings of right-angled Artin group A 1 on the complement of a linear forest into another right-angled Artin group A 2 can be reduced to full embeddings of the defining graph of A 1 into the extension graph of the defining graph of A 2 .

The RAAGs on the complement graphs of path graphs in mapping class groups

“…Then, by Baudisch's theorem [2, Theorem 1.2], ψ(u) M , ψ(v) M is isomorphic to an abelian group or free group of rank 2. Since the Dehn twist [T v,j ] do not appear in the presentation of ψ(u), and since [T v,j ] M e(v,j) is not commutative with ψ(u) M , it follows that ψ(u) M and ψ(v) M are not commutative in G(X T ) (see [13,Lemma 2.2]). Therefore ψ(u) M , ψ(v) M must be a free group of rank 2.…”

“…Note that the vertices v 1 and v 3 are commutative in G(Γ ), because the vertices u 1 and u 3 are commutative in G(ι(P 3 )) (see [13,Lemma 2.2]). Hence, the endpoints v 1 and v 3 of P do not span an edge in Γ , and therefore (v 1 , v 2 , v 3 ) induces P 3 in Γ .…”