Embedding products of trees into higher rank

I will present a joint work with Thang Nguyen where we show that there exists a quasi-isometric embedding of the product of n copies of the hyperbolic plane into any symmetric space of non-compact type of rank n, and there exists a bi-Lipschitz embedding of the product of n copies of the 3-regular tree into any thick Euclidean building of rank n. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings. I will start by describing the objects and the embeddings, and then give a detailed sketch of the proof in rank 2.