RAAGs in Diffeos

Abstract

We survey embeddability results related to RAAGs (right-angled Artin groups) and various automorphism groups of manifolds. We give two different methods of embedding a RAAG to into another, and deduce that every RAAG embeds into some braid groups. This gives the unsolvability of the isomorphism problem for finitely presented subgroups of braid groups. Also, we prove that every RAAG is a quasi-isometrically embedded subgroup of the symplectomorphism groups of the disk and the sphere, given with suitable $L^p$ metrics. Finally, we embed RAAGs in the smooth diffeomorphism group of the real line. These results reveal many closed hyperbolic manifold subgroups of diffeomorphism groups of manifolds.