Geometry & Topology

Quasi-isometric embeddings of symmetric spaces

David Fisher and Kevin Whyte

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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

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

Received: 24 May 2017
Revised: 14 January 2018
Accepted: 4 March 2018
First available in Project Euclid: 26 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 53C24: Rigidity results 53C35: Symmetric spaces [See also 32M15, 57T15]

symmetric spaces quasi-isometries coarse geometry rigidity


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.

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