Geometry & Topology
- Geom. Topol.
- Volume 17, Number 1 (2013), 493-530.
Embedability between right-angled Artin groups
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.
Geom. Topol., Volume 17, Number 1 (2013), 493-530.
Received: 20 March 2012
Accepted: 20 November 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F36: Braid groups; Artin groups
Kim, Sang-hyun; Koberda, Thomas. Embedability between right-angled Artin groups. Geom. Topol. 17 (2013), no. 1, 493--530. doi:10.2140/gt.2013.17.493. https://projecteuclid.org/euclid.gt/1513732530