Innovations in Incidence Geometry

Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

Pierre-Emmanuel Caprace

Full-text: Open access


Let (W,S) be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of (W,S) for W to be relatively hyperbolic with respect to a collection of finitely generated subgroups. The peripheral subgroups are necessarily parabolic subgroups (in the sense of Coxeter group theory). As an application, we present a criterion for the maximal flats of the Davis complex of (W,S) to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type (W,S) are isolated.

Article information

Innov. Incidence Geom., Volume 10, Number 1 (2009), 15-31.

Received: 26 September 2007
Accepted: 30 May 2009
First available in Project Euclid: 28 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 20F67: Hyperbolic groups and nonpositively curved groups 20F69: Asymptotic properties of groups

Coxeter group building isolated flat relative hyperbolicity


Caprace, Pierre-Emmanuel. Buildings with isolated subspaces and relatively hyperbolic Coxeter groups. Innov. Incidence Geom. 10 (2009), no. 1, 15--31. doi:10.2140/iig.2009.10.15.

Export citation


  • B. Brink and R. B. Howlett, A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296 (1993), 179–190.
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.
  • M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997), 245–268.
  • B. H. Bowditch, Relatively hyperbolic groups, preprint, available at, 1999.
  • P.-E. Caprace, Conjugacy of $2$-spherical subgroups of Coxeter groups and parallel walls, Algebr. Geom. Topol. 6 (2006), 1987–2029.
  • P.-E. Caprace and K. Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (2010), 1296–1319.
  • P.-E. Caprace and F. Haglund, On geometric flats in the ${\rm CAT}(0)$ realization of Coxeter groups and Tits buildings, Canad. J. Math. 61 (2009), 740–761.
  • P.-E. Caprace and B. Mühlherr, Reflection triangles in Coxeter groups and biautomaticity, J. Group Theory 8 (2005), 467–489.
  • M. Davis, Buildings are ${\rm CAT}(0)$, in Geometry and cohomology in group theory (Durham, 1994) (Cambridge) (H. Kropholler et al., ed.), London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press, 1998, pp. 108–123.
  • V. V. Deodhar, On the root system of a Coxeter group, Comm. Algebra 10 (1982), 611–630.
  • ––––, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel) 53 (1989), 543–546.
  • C. Drutu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), 959–1058.
  • B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), 810–840.
  • G. C. Hruska and B. Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005), 1501–1538.
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., vol. 29, Cambridge University Press, Cambridge, 1990.
  • R. Kangaslampi and A. Vdovina, Triangular hyperbolic buildings, C. R. Math. Acad. Sci. Paris 342 (2006), 125–128.
  • B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 115–197.
  • D. Krammer, The conjugacy problem for Coxeter groups, Groups Geom. Dyn. 3 (2009), 71–171.
  • G. A. Margulis and \bf È. B. Vinberg, Some linear groups virtually having a free quotient, J. Lie Theory 10 (2000), 171–180.
  • G. Moussong, Hyperbolic Coxeter Groups, Ph.D. thesis, Ohio State University, 1988.
  • D. Y. Rebbechi, Algorithmic Properties of Relatively Hyperbolic Groups, Ph.D. thesis, Rutgers University, 2001, family arXiv:math.GR/0302245.
  • J. Swiatkowski, Regular path systems and (bi)automatic groups, Geom. Dedicata 118 (2006), 23–48.
  • J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math., Vol. 386, Springer-Verlag, Berlin, 1974.