Coarse embeddings of products of trees as quasi-isometry invariants

We consider the maximal number of factors of a product of bushy trees that can be quasi-isometrically, or even coarsely embedded into various groups of interest, including mapping class groups, Torelli groups, Johnson kernels, surface braid groups, and Bestvina-Brady groups. We use this to quasi-isometrically distinguish groups from the above classes, and also to rule out coarse embeddings between them. All these are applications of general statements about coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.