Introduced by Gromov in the 80’s, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we are particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is well understood, and it is well known, by results of Anderson–Schroeder and Kleiner, that the rank of these spaces is monotone under quasi-isometric embeddings. This is no longer the case for coarse embeddings as shown by horospherical embeddings. However, we show that in the absence of a Euclidean factor in the domain, the rank is monotone under coarse embeddings. This answers a question raised by Fisher and Whyte. The same conclusion holds when the target space is replaced by a proper cocompact CAT(0) space or a mapping class group. In the case of symmetric spaces and Euclidean buildings, we can further relax the condition on the domain by allowing it to contain a Euclidean factor of dimension 1, thereby answering a question of Gromov.
Oussama Bensaid (Sat,) studied this question.