Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings

Antoine Clais 1
  • 1 Laboratoire Paul Painlevé, Université Lille 1, 59655 Villeneuve d’Ascq, France


In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D.Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity of right-angled Fuchsian buildings.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Peter Abramenko and Kenneth S. Brown. Buildings, volume 248 of Graduate Texts in Mathematics. Springer, New York, 2008. Theory and applications.

  • [2] Mario Bonk and Bruce Kleiner. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math., 150(1):127–183, 2002.

  • [3] Mario Bonk and Bruce Kleiner. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geom. Topol., 9:219–246, 2005.

  • [4] Marc Bourdon and Bruce Kleiner. Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups. Groups Geom. Dyn., 7(1):39–107, 2013.

  • [5] M. Bourdon. Immeubles hyperboliques, dimension conforme et rigidité de Mostow. Geom. Funct. Anal., 7(2):245–268, 1997.

  • [6] Brian H. Bowditch. Cut points and canonical splittings of hyperbolic groups. Acta Math., 180(2):145–186, 1998.

  • [7] Marc Bourdon and Hervé Pajot. Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. Proc. Amer. Math. Soc., 127(8):2315–2324, 1999.

  • [8] Marc Bourdon and Hervé Pajot. Rigidity of quasi-isometries for some hyperbolic buildings. Comment. Math. Helv., 75(4):701–736, 2000.

  • [9] James W. Cannon. The combinatorial Riemann mapping theorem. Acta Math., 173(2):155–234, 1994.

  • [10] Pierre-Emmanuel Caprace. Automorphism groups of right-angled buildings: simplicity and local splittings. Fund. Math., 224(1):17–51, 2014.

  • [11] Matias Carrasco. Thése de doctorat : Jauge conforme des espaces métrique compacts. Université de Provence, 2011.

  • [12] M. Coornaert, T. Delzant, and A. Papadopoulos. Géométrie et théorie des groupes, volume 1441 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1990. Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups],With an English summary.

  • [13] Ruth Charney. An introduction to right-angled Artin groups. Geom. Dedicata, 125:141–158, 2007.

  • [14] A. Clais. Parallel residues in buildings admitting a group action. Pre-print arXiv:1312.5541, 2013.

  • [15] H. S. M. Coxeter. Regular polytopes. Dover Publications, Inc., New York, third edition, 1973.

  • [16] J. W. Cannon and E. L. Swenson. Recognizing constant curvature discrete groups in dimension 3. Trans. Amer. Math. Soc., 350(2):809–849, 1998.

  • [17] Michael W. Davis. A hyperbolic 4-manifold. Proc. Amer. Math. Soc., 93(2):325–328, 1985.

  • [18] MichaelW. Davis. Buildings are CAT(0). In Geometry and cohomology in group theory (Durham, 1994), volume252 of London Math. Soc. Lecture Note Ser., pages 108–123. Cambridge Univ. Press, Cambridge, 1998.

  • [19] Michael W. Davis. The geometry and topology of Coxeter groups, volume 32 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2008.

  • [20] MichaelW. Davis and John Meier. The topology at infinity of Coxeter groups and buildings. Comment.Math. Helv., 77(4):746– 766, 2002.

  • [21] Jan Dymara and Damian Osajda. Boundaries of right-angled hyperbolic buildings. Fund. Math., 197:123–165, 2007.

  • [22] Étienne Ghys and Pierre de la Harpe. Espaces métriques hyperboliques. In Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988), volume 83 of Progr. Math., pages 27–45. Birkhäuser Boston, Boston, MA, 1990.

  • [23] Damien Gaboriau and Frédéric Paulin. Sur les immeubles hyperboliques. Geom. Dedicata, 88(1-3):153–197, 2001.

  • [24] Peter Haïssinsky. Empilements de cercles et modules combinatoires. Ann. Inst. Fourier (Grenoble), 59(6):2175–2222, 2009.

  • [25] Peter Haïssinsky. Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités [d’après Mostow, Pansu, Bourdon, Pajot, Bonk, Kleiner. . .]. Astérisque, (326):Exp. No. 993, ix, 321–362 (2010), 2009. Séminaire Bourbaki. Vol. 2007/2008.

  • [26] Juha Heinonen. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.

  • [27] Juha Heinonen and Pekka Koskela. Quasiconformal maps in metric spaces with controlled geometry. Acta Math., 181(1):1– 61, 1998.

  • [28] Frédéric Haglund and Frédéric Paulin. Constructions arborescentes d’immeubles. Math. Ann., 325(1):137–164, 2003.

  • [29] Ilya Kapovich and Nadia Benakli. Boundaries of hyperbolic groups. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), volume 296 of Contemp. Math., pages 39–93. Amer. Math. Soc., Providence, RI, 2002.

  • [30] Stephen Keith and Bruce Kleiner. In preparation.

  • [31] S. Keith and T. Laakso. Conformal Assouad dimension and modulus. Geom. Funct. Anal., 14(6):1278–1321, 2004.

  • [32] Bruce Kleiner. The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity. In International Congress of Mathematicians. Vol. II, pages 743–768. Eur. Math. Soc., Zürich, 2006.

  • [33] Charles Loewner. On the conformal capacity in space. J. Math. Mech., 8:411–414, 1959.

  • [34] John Meier. When is the graph product of hyperbolic groups hyperbolic? Geom. Dedicata, 61(1):29–41, 1996.

  • [35] John M.Mackay and Jeremy T. Tyson. Conformal dimension, volume 54 of University Lecture Series. AmericanMathematical Society, Providence, RI, 2010. Theory and application.

  • [36] James R. Munkres. Topology: a first course. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975.

  • [37] Pierre Pansu. Dimension conforme et sphère à l’infini des variétés à courbure négative. Ann. Acad. Sci. Fenn. Ser. A IMath., 14(2):177–212, 1989.

  • [38] Mark Ronan. Lectures on buildings, volume 7 of Perspectives in Mathematics. Academic Press, Inc., Boston, MA, 1989.

  • [39] Dennis Sullivan. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. (N.S.), 6(1):57–73, 1982.

  • [40] Jacques Tits. Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, Vol. 386. Springer-Verlag, Berlin-New York, 1974.

  • [41] Jeremy Tyson. Quasiconformality and quasisymmetry in metricmeasure spaces. Ann. Acad. Sci. Fenn.Math., 23(2):525–548, 1998.

  • [42] Jussi Väisälä. Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics, Vol. 229. Springer- Verlag, Berlin-New York, 1971.

  • [43] Jussi Väisälä. Quasi-Möbius maps. J. Analyse Math., 44:218–234, 1984/85.

  • [44] Matti Vuorinen. Conformal geometry and quasiregular mappings, volume 1319 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, 1988.


Journal + Issues