Embedding products of trees into higher rank

We show that there exists a quasi‐isometric embedding of the product of copies of into any symmetric space of non‐compact type of rank , and there exists a bi‐Lipschitz embedding of the product of copies of the 3‐regular tree into any thick Euclidean building of rank with co‐compact affine Weyl group. This extends a previous result of Fisher–Whyte. The proof is purely geometrical, and the result also applies to the non–Bruhat–Tits buildings.