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We show that there exists a quasi-isometric embedding of the product of n copies of Hₑ² 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 T₃ into any thick Euclidean building of rank n 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.
Bensaid et al. (Fri,) studied this question.