Volume 17, issue 1 (2013)

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Embedability between right-angled Artin groups

Sang-hyun Kim and Thomas Koberda

Geometry & Topology 17 (2013) 493–530

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γe of Γ. We produce a second graph Γke, the clique graph of Γe, by adding an extra vertex for each complete subgraph of Γe. We prove that each finite induced subgraph Λ of Γe gives rise to an inclusion A(Λ) A(Γ). Conversely, we show that if there is an inclusion A(Λ) A(Γ) then Λ is an induced subgraph of Γke. These results have a number of corollaries. Let P4 denote the path on four vertices and let Cn denote the cycle of length n. We prove that A(P4) embeds in A(Γ) if and only if P4 is an induced subgraph of Γ. We prove that if F is any finite forest then A(F) embeds in A(P4). We recover the first author’s result on co-contraction of graphs, and prove that if Γ has no triangles and A(Γ) contains a copy of A(Cn) for some n 5, then Γ contains a copy of Cm for some 5 m n. We also recover Kambites’ Theorem, which asserts that if A(C4) embeds in A(Γ) then Γ contains an induced square. We show that whenever Γ is triangle-free and A(Λ) < A(Γ) then there is an undistorted copy of A(Λ) in A(Γ). Finally, we determine precisely when there is an inclusion A(Cm) A(Cn) and show that there is no “universal” two–dimensional right-angled Artin group.

right-angled Artin group, mapping class group, surface group, co-contraction
Mathematical Subject Classification 2010
Primary: 20F36
Received: 20 March 2012
Accepted: 20 November 2012
Published: 2 April 2013
Proposed: Benson Farb
Seconded: Cameron Gordon, Danny Calegari
Sang-hyun Kim
Department of Mathematical Sciences
Daejeon 305-701
Republic of Korea
Thomas Koberda
Department of Mathematics
Yale University
P.O. Box 208283
New Haven, CT 06511