Quasiflats with holes in reductive groups

Wortman, Kevin

doi:10.2140/agt.2006.6.91

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Quasiflats with holes in reductive groups

Kevin Wortman

Algebraic & Geometric Topology 6 (2006) 91–117

DOI: 10.2140/agt.2006.6.91

arXiv: math.GT/0401360

Abstract

We give a new proof of a theorem of Kleiner–Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the rank of the Euclidean space is not less than the rank of the target. A bound on the size of the neighborhood and on the number of flats is determined by the size of the quasi-isometry constants.

Without using asymptotic cones, our proof focuses on the intrinsic geometry of symmetric spaces and Euclidean buildings by extending the proof of Eskin–Farb’s quasiflat with holes theorem for symmetric spaces with no Euclidean factors.

Keywords

Euclidean building, symmetric space, geometric realization of algebraic 2 complexes, quasi-isometry

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 20G30, 22E40

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Publication

Received: 18 November 2004

Accepted: 11 August 2005

Published: 24 February 2006

Authors

Kevin Wortman
Department of Mathematics Yale University 10 Hillhouse Ave PO Box 208283 New Haven CT 06520-8283 USA

Volume 6, issue 1 (2006)