Algebraic & Geometric Topology

Quasiflats with holes in reductive groups

Kevin Wortman

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796507

Digital Object Identifier
doi:10.2140/agt.2006.6.91

Mathematical Reviews number (MathSciNet)
MR2199455

Zentralblatt MATH identifier
1166.20038

Subjects
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]

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

Citation

Wortman, Kevin. Quasiflats with holes in reductive groups. Algebr. Geom. Topol. 6 (2006), no. 1, 91--117. doi:10.2140/agt.2006.6.91. https://projecteuclid.org/euclid.agt/1513796507


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References

  • K S Brown, Buildings, Springer, New York (1989)
  • A Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces, J. Amer. Math. Soc. 11 (1998) 321–361
  • A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653–692
  • A Eskin, B Farb, Quasi-flats in $H\sp 2\times H\sp 2$, from: “Lie groups and ergodic theory (Mumbai, 1996)”, Tata Inst. Fund. Res. Stud. Math. 14, Tata Inst. Fund. Res., Bombay (1998) 75–103
  • B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115–197 (1998)
  • H M Morse, }, Trans. Amer. Math. Soc. 26 (1924) 25–60
  • G D Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J. (1973)
  • P Pansu, }, Ann. of Math. $(2)$ 129 (1989) 1–60
  • K Wortman, Quasi-isometric rigidity of higher rank $S$-arithmetic lattices, preprint
  • K Wortman, Quasi-isometries of rank one $S$-arithmetic lattices, preprint