Anti-trees and right-angled Artin subgroups of braid groups

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Written by Sang-hyun Kim and Thomas Koberda. link

Background
  1. A RAAG (right-angled Artin group) on a graph X is defined as \(G(X) = \langle v\in V(X) \mid [a,b]=1\text{ if }\{a,b\}\not\in E(X)\rangle\). Note this definition is opposite to a usual convention for RAAGs.
  2. M. Kapovich proved that each RAAG (group theoretically) injects into the symplectomorphism group of the sphere.
  3. A qi group embedding between groups with metrics is an injective group homomorphism which is a quasi-isometry as well.
Main result

Each RAAG admits a qi group embedding into the following groups:

  1. some RAAG defined by a tree.
  2. some braid group.
  3. the symplectomorphism group of the disk with Lp metric for p ≥ 1.
  4. the symplectomorphism group of the sphere with Lp metric for p > 2.