Algebraic & Geometric Topology

Quasiflats with holes in reductive groups

Kevin Wortman

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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.

Article information

Algebr. Geom. Topol., Volume 6, Number 1 (2006), 91-117.

Received: 18 November 2004
Accepted: 11 August 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20G30: Linear algebraic groups over global fields and their integers 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

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


Wortman, Kevin. Quasiflats with holes in reductive groups. Algebr. Geom. Topol. 6 (2006), no. 1, 91--117. doi:10.2140/agt.2006.6.91.

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