Path lifting properties and embedding between RAAGs

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• 作者：Eon-Kyung Lee^a ; ^{eonkyung@sejong.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Sang-Jin Lee^b ; ^{sangjin@konkuk.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 20F36 ; secondary ; 20F65
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：448
• 期：Complete
• 页码：575-594
• 全文大小：542 K

文摘

For a finite simplicial graph Γ, let G(Γ)$G(\mathrm{\Gamma })$ denote the right-angled Artin group on the complement graph of Γ. In this article, we introduce the notions of “induced path lifting property” and “semi-induced path lifting property” for immersions between graphs, and obtain graph theoretical criteria for the embeddability between right-angled Artin groups. We recover the result of S.-h. Kim and T. Koberda that an arbitrary G(Γ)$G(\mathrm{\Gamma })$ admits a quasi-isometric group embedding into G(T)$G(T)$ for some finite tree T. The upper bound on the number of vertices of T   is improved from 34b2f186807eea46e7d" title="Click to view the MathML source">2^2(m−1)2$2^{2^{{(m-1)}^2}}$ to m2^m−1$m{2}^{m-1}$, where m is the number of vertices of Γ. We also show that the upper bound on the number of vertices of T   is at least 2^m/4$2^{m/4}$. Lastly, we show that b4001ab6617791908774954" title="Click to view the MathML source">G(C_m)$G(C_m)$ embeds in G(P_n)$G(P_n)$ for n⩾2m−2$n⩾2m-2$, where C_m$C_m$ and P_n$P_n$ denote the cycle and path graphs on m and n vertices, respectively.