Embedability between right-angled Artin groups
By Sang-hyun Kim, Thomas Koberda. 2011
122 citations recorded by the source corpus.
In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph A , we produce a new graph through a purely combinatorial procedure, and call it the extension graph A e of A . We produce a second graph A e k , the clique graph of A e , by adding an extra vertex for each complete subgraph of A e . We prove that each finite induced subgraph E of A e gives rise to an inclusion A.E/! A.A/. Conversely, we show that if there is an inclusion A.E/! A.A/ then E is an induced subgraph of A e k . 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.A/ if and only if P4 is an induced subgraph of A . 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 A has no triangles and A.A/ contains a copy of A.Cn/ for some n 5, then A contains a copy of Cm for some 5 m n. We also recover Kambites’ Theorem, which asserts that if A.C4/ embeds in A.A/ then A contains an induced square. We show that whenever A is triangle-free and A.E/ < A.A/ then there is an undistorted copy of A.E/ in A.A/. 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. 20F36
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/837b2406a9133848fe5e481fa315cbad7e70d102
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Citations in the corpus
- Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (2010, background): “n system is not simply connected (see [5], also [19]). A systematic study of right-angled Artin subgroups of a given right-angled Artin group has recently been carried out by S. Kim and the author in [37]. 1.4. Commensurability questions. Recall that two groups Gand Hare commensurable if there exist”