Quasi-isometric embeddings of symmetric spaces

Fisher, David¹ ; Whyte, Kevin²

¹ Department of Mathematics, Indiana University, Bloomington, IN, United States
² Department of Mathematics, University of Illinois at Chicago, Chicago, IL, United States

Geometry & topology, Tome 22 (2018) no. 5, pp. 3049-3082.

Résumé

This paper opens the study of quasi-isometric embeddings of symmetric spaces. The main focus is on the case of equal and higher rank. In this context some expected rigidity survives, but some surprising examples also exist. In particular there exist quasi-isometric embeddings between spaces $X$ and $Y$ where there is no isometric embedding of $X$ into $Y$. 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 quasiflat theorem, which says that the maximal quasiflat is within bounded distance of a finite union of flats. We improve this by showing that the quasiflat is in fact flat off of a subset of codimension $2$.

Détail

DOI : 10.2140/gt.2018.22.3049

Classification : 22E40, 53C24, 53C35
Keywords: symmetric spaces, quasi-isometries, coarse geometry, rigidity

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