Abstract
In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph , we produce a new graph through a purely combinatorial procedure, and call it the extension graph of . We produce a second graph , the clique graph of , by adding an extra vertex for each complete subgraph of . We prove that each finite induced subgraph of gives rise to an inclusion . Conversely, we show that if there is an inclusion then is an induced subgraph of . These results have a number of corollaries. Let denote the path on four vertices and let denote the cycle of length . We prove that embeds in if and only if is an induced subgraph of . We prove that if is any finite forest then embeds in . We recover the first author’s result on co-contraction of graphs, and prove that if has no triangles and contains a copy of for some , then contains a copy of for some . We also recover Kambites’ Theorem, which asserts that if embeds in then contains an induced square. We show that whenever is triangle-free and then there is an undistorted copy of in . Finally, we determine precisely when there is an inclusion and show that there is no “universal” two–dimensional right-angled Artin group.
Citation
Sang-hyun Kim. Thomas Koberda. "Embedability between right-angled Artin groups." Geom. Topol. 17 (1) 493 - 530, 2013. https://doi.org/10.2140/gt.2013.17.493
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