Embeddability of right-angled Artin groups on complements of trees

Abstract

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 Γ e for Γ, and established the Extension Graph Theorem: for finite simplicial graphs Γ 1 and Γ 2 if Γ 1 embeds into Γ e 2 as an induced subgraph then A(Γ 1 ) embeds into A(Γ 2 ). In this article we show that the converse of this theorem does not hold for the case Γ 1 is the complement of a tree and for the case Γ 2 is the complement of a path graph.

A Study on Research Trend of Thesis and Dissertation in the Department Korean Traditional music at Seoul National University

“…Extension graphs are usually infinite and locally infinite. They are very useful in the study of right-angled Artin groups such as the embeddability problem between right-angled Artin groups [KK13,KK14a,LL16,LL18]. Let d e denote the graph metric d Γ e .…”

Actions of right-angled Artin groups in low dimensions

“…The question of which rightangled Artin groups occur as subgroups of ApΓq was first systematically studied by Kim and the author [63,64,66], cf. [24,77]. We outline some of the main features of this theory.…”

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.…”