Right-angled Artin groups and Out(${\mathbb{F}}_n$) – I. Quasi-isometric embeddings

Abstract

We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are modeled on the homomorphisms into the mapping class group constructed by Clay, Leininger, and Mangahas. Toward this goal, we develop tools in the free group setting that mirror those for surface groups and discuss various analogs of subsurface projection.