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

From w

Written by Sang-hyun Kim and Thomas Koberda. link

- Background

- 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. - M. Kapovich proved that each RAAG (group theoretically) injects into the symplectomorphism group of the sphere.
- 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:

- some RAAG defined by a tree.
- some braid group.
- the symplectomorphism group of the disk with L
^{p}metric for p ≥ 1. - the symplectomorphism group of the sphere with L
^{p}metric for p > 2.