Coarse embeddings, introduced by Gromov in the 80s, are generalizations of quasi-equal length embeddings when the control function is not necessarily affine. In this paper, we are particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to symmetric space and high-rank building stiffness results by Anderson-Schroeder, Kleiner, Kleiner-Leeb, Eskin-Farb, and Fisher-Whyte. In particular, it is well known that the rank of these spaces is monotonic under quasi-isometric embeddings. This is not the case for the coarse embeddings shown in Holosphere embeddings. However, if the domain has no Euclidean factor, we show that the ranks are monotonic under the coarse embedding. This is an answer to a question by David Fisher and Kevin Whyte. This is true even if we replace the target space with an appropriate cocompact CAT(0) space or mapping class group. Between the symmetric space and the Euclidean building, we can also relax the region condition by including a one-dimensional Euclidean factor in the region, answering Gromov’s question.
