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Quasi-isometric co-Hopficity of non-uniform lattices in rank-one semi-simple Lie groups


Authors: Ilya Kapovich and Anton Lukyanenko
Journal: Conform. Geom. Dyn. 16 (2012), 269-282
MSC (2010): Primary 20F65; Secondary 53C23
DOI: https://doi.org/10.1090/S1088-4173-2012-00246-9
Published electronically: October 15, 2012
MathSciNet review: 2983835
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Abstract: We prove that if $ G$ is a non-uniform lattice in a rank-one semi-simple Lie group $ \ne$$ \text {Isom}( \mathbb{H}^2_{\mathbb{R}})$, then $ G$ is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding $ G\to G$ is coarsely surjective and thus is a quasi-isometry.


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Additional Information

Ilya Kapovich
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: kapovich@math.uiuc.edu

Anton Lukyanenko
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: anton@lukyanenko.net

DOI: https://doi.org/10.1090/S1088-4173-2012-00246-9
Received by editor(s): April 17, 2012
Published electronically: October 15, 2012
Additional Notes: The first author was supported by the NSF grant DMS-0904200.
The authors acknowledge support from the National Science Foundation grant DMS-1107452 “RNMS: Geometric structures and representation varieties”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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