Coarse embeddings of symmetric spaces and Euclidean buildings

TL;DR

This paper investigates coarse embeddings between symmetric spaces and Euclidean buildings, establishing conditions under which the rank is preserved, and extends results to broader classes of target spaces, answering longstanding questions in geometric group theory.

Contribution

It proves that rank monotonicity holds for coarse embeddings without Euclidean factors in the domain, extending known rigidity results and answering questions by Fisher, Whyte, and Gromov.

Findings

Rank is monotonous under coarse embeddings without Euclidean factors in the domain.

The result extends to proper cocompact CAT(0) spaces and mapping class groups.

Allows Euclidean factors of dimension 1 in the domain for certain embeddings.

Abstract

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 will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb, Eskin-Farb and Fisher-Whyte. In particular, it is well known that the rank of these spaces is monotonous 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 monotonous under coarse embeddings. This answers a question by David Fisher and Kevin Whyte. This still holds when we replace the target space by a proper cocompact CAT(0) space or by a mapping class group. Between symmetric spaces and Euclidean buildings, we can also relax the condition on the domain by allowing it to contain a Euclidean factor of dimension 1, answering a question by Gromov.