Right-angled Artin groups and full subgraphs of graphs

Abstract

For a finite graph Γ, let G(Γ) be the right-angled Artin group defined by the complement graph of Γ. We show that, for any linear forest Λ and any finite graph Γ, G(Λ) can be embedded into G(Γ) if and only if Λ can be realised as a full subgraph of Γ. We also prove that if we drop the assumption that Λ is a linear forest, then the above assertion does not hold, namely, for any finite graph Λ, which is not a linear forest, there exists a finite graph Γ such that G(Λ) can be embedded into G(Γ), though Λ cannot be embedded into Γ as a full subgraph.2010 Mathematics Subject Classification. 20F36 (primary).

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

“…However, E. Lee and S. Lee [5] pointed out that the above "Theorem" is incorrect by giving a counter-example. Thus the author's proof of Theorem 1.1 in [3] is not valid.…”

“…The complement Λ c of a graph Λ is the graph consisting of the vertex set V (Λ c ) = V (Λ) and the edge set E(Λ c ) = {{u, v} | u, v ∈ V (Λ), {u, v} / ∈ E(Λ)}. In [3] the author "proved" the following theorem. Theorem 1.1.…”

“…We remark that Theorem 1.1 is equivalent to [3, Theorem 1.3 (1)]. In the proof of [3, Theorem 1.3 (1)], the author used the following "Theorem" (see the second line of the proof of Theorem 3.6 in [3]). Theorem 3.14]).…”

“…In fact, the author applied "Theorem 1.2" only for the complement of linear forests in the proof of Theorem 1.3(1) in [3].…”

Embeddability of right-angled Artin groups on complements of trees

“…Kim and Koberda showed that the family of finite trees provides a universal family of graphs for right-angled Artin groups. Using the fact that if Γ 1 is an edge-contraction of Γ 2 then G(Γ 1 ) G(Γ 2 ) [Kim08, KK13], we can see that the family of finite trees with degree 3 at each vertex is also universal [Kat16]. It would be interesting to find a universal family smaller than this.…”