## Geometry & Topology

### Quasi-isometric embeddings of symmetric spaces

#### Abstract

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 $2$.

#### Article information

Source
Geom. Topol., Volume 22, Number 5 (2018), 3049-3082.

Dates
Revised: 14 January 2018
Accepted: 4 March 2018
First available in Project Euclid: 26 March 2019

https://projecteuclid.org/euclid.gt/1553565678

Digital Object Identifier
doi:10.2140/gt.2018.22.3049

Mathematical Reviews number (MathSciNet)
MR3811777

Zentralblatt MATH identifier
1392.22004

#### Citation

Fisher, David; Whyte, Kevin. Quasi-isometric embeddings of symmetric spaces. Geom. Topol. 22 (2018), no. 5, 3049--3082. doi:10.2140/gt.2018.22.3049. https://projecteuclid.org/euclid.gt/1553565678

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