Quasi-Isometric Embeddings of Symmetric Spaces

TL;DR

This paper establishes a rigidity theorem for quasi-isometric embeddings of higher rank symmetric spaces, showing they are close to isometric embeddings, and provides new examples of such embeddings, including cases where no isometric embeddings exist.

Contribution

The paper proves a new rigidity theorem for quasi-isometric embeddings of symmetric spaces and constructs surprising examples of embeddings where isometric ones do not exist.

Findings

Quasi-isometric embeddings of equal rank symmetric spaces are close to isometric embeddings.

Constructed embeddings of SL(n,R) into Sp(2(n-1),R) where no isometric embeddings are possible.

Improved the understanding of quasi-flat structures by showing they are flat off a codimension 2 subset.

Abstract

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,\mathbb R)$ into $Sp(2(n-1),\mathbb R)$ when no isometric embeddings exist. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that maximal quasi-flat is within bounded distance of a finite union of flats. We improve this by showing that the quasi-flat is in fact flat off of a subset of codimension $2$.