Geodesic
Quasiflats with holes in reductive groups
Wortman, Kevin1
1Department of Mathematics, Yale University, 10 Hillhouse Ave, PO Box 208283, New Haven CT 06520-8283, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 91-117
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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.

DOI : 10.2140/agt.2006.6.91
Keywords: Euclidean building, symmetric space, geometric realization of algebraic 2 complexes, quasi-isometry

Wortman, Kevin  1

1 Department of Mathematics, Yale University, 10 Hillhouse Ave, PO Box 208283, New Haven CT 06520-8283, USA 
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Wortman, Kevin. Quasiflats with holes in reductive groups. Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 91-117. doi: 10.2140/agt.2006.6.91

[1] K S Brown, Buildings, Springer (1989)

[2] A Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces, J. Amer. Math. Soc. 11 (1998) 321

[3] A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653

[4] A Eskin, B Farb, Quasi-flats in $H^2\times H^2$, from: "Lie groups and ergodic theory (Mumbai, 1996)", Tata Inst. Fund. Res. Stud. Math. 14, Tata Inst. Fund. Res. (1998) 75

[5] B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997)

[6] H M Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924) 25

[7] G D Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press (1973)

[8] P Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. $(2)$ 129 (1989) 1

[9] K Wortman, Quasi-isometric rigidity of higher rank $S$–arithmetic lattices, preprint

[10] K Wortman, Quasi-isometries of rank one $S$–arithmetic lattices, preprint

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