Volume 6, issue 1 (2006)

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Quasiflats with holes in reductive groups

Kevin Wortman

Algebraic & Geometric Topology 6 (2006) 91–117

arXiv: math.GT/0401360


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.

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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Received: 18 November 2004
Accepted: 11 August 2005
Published: 24 February 2006
Kevin Wortman
Department of Mathematics
Yale University
10 Hillhouse Ave
PO Box 208283
New Haven CT 06520-8283