Geometry & Topology

Metric characterizations of spherical and Euclidean buildings

Ruth Charney and Alexander Lytchak

Full-text: Open access

Abstract

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties.

Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 521-550.

Dates
Received: 23 November 2000
Revised: 11 May 2001
Accepted: 18 May 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883036

Digital Object Identifier
doi:10.2140/gt.2001.5.521

Mathematical Reviews number (MathSciNet)
MR1833752

Zentralblatt MATH identifier
1002.51008

Subjects
Primary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
buildings CAT(0) spaces spherical buildings Euclidean buildings metric characterisation

Citation

Charney, Ruth; Lytchak, Alexander. Metric characterizations of spherical and Euclidean buildings. Geom. Topol. 5 (2001), no. 2, 521--550. doi:10.2140/gt.2001.5.521. https://projecteuclid.org/euclid.gt/1513883036


Export citation

References

  • W Ballmann Lectures on Spaces of Nonpositive Curvature, DMV Seminar 25, Birkhäuser (1995)
  • W Ballmann M Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. 82 (1996) 169–209
  • W Ballmann M Brin, Diameter rigidity of spherical polyhedra, Duke Math. J. 97 (1999) 235–259
  • M Bridson A Haefliger Metric Spaces of Non-positive Curvature Springer–Verlag (1999)
  • N Bourbaki Groupes et Algebrès de Lie, Ch. 4–6, Masson, Paris (1981)
  • K Brown Buildings Springer–Verlag, New York (1989)
  • R Charney M W Davis, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer. J. Math. 115 (1993) 929–1009
  • M W Davis, Buildings are CAT(0), from Geometry and Cohomology in Group Theory, ed. by P Kropholler, G Niblo and R Stöhr, LMS Lecture Notes Series 252, Cambridge University Press (1998)
  • M Gromov, Hyperbolic groups, from Essays in Group Theory, ed. by S M Gersten, MSRI Publ. 8, Springer–Verlag, New York (1987) 75–264
  • W M Kantor, Generalized polygons, SCAB's and GAB's, from Buildings and the Geometry of Diagrams, LNM 1181, Springer–Verlag, New York (1986) 79–158
  • B Kleiner B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. IHES 86 (1997) 115–197
  • A Neumaier, Some aporadic geometries related to PG(3,2), Arch. Math. 42 (1984) 89–96
  • M Ronan Lectures on Buildings, Perspectives in Mathematics, Vol. 7, Academic Press, San Diego (1989)
  • J Tits, A local approach to buildings, from The Geometric Vein, Coxeter Festschrift, ed. by C Davis, B Grüenbaum and F A Sherk, Springer–Verlag, New York (1981) 519–547 s