Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T10:54:51.768Z Has data issue: false hasContentIssue false
  • English
  • Français

On the tree-likeness of hyperbolic spaces

Published online by Cambridge University Press:  10 April 2017

MATTHIAS HAMANN
MATTHIAS HAMANN*
Affiliation:
Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. e-mail: matthias.hamann@math.uni-hamburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Inside any proper hyperbolic geodesic space X we construct a rooted topological ${\mathbb R}$-tree T that reflects the geometry of X in the following sense. All rays in T are quasi-geodesic in X. Every geodesic ray in X lies eventually close to a ray of T. The embedding of T in X extends continuously to their boundaries in a finite-to-one way, the number of boundary points of T mapping to a given boundary point of X being bounded if the (Assouad) dimension of the boundary of X is finite.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 345 - 361
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Alonso, J.M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M. and Short, H. Notes on word hyperbolic groups. Group Theory from a Geometrical Viewpoint (Trieste, 1990) (Ghys, E., Haefliger, A. and Verjovsky, A., eds.) (World Scientific, 1991), pp. 3–63.Google Scholar
[2] Assouad, P. Plongements lipschitziens dans $\mathbb {R}$ n . Bull. Soc. Math. France 111 (1983), no. 4, 429–448.Google Scholar
[3] Bell, G. and Dranishnikov, A. Asymptotic dimension. Topology Appl. 155 (2008), no. 12, 1265–1296.Google Scholar
[4] Benjamini, I. and Schramm, O. Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. 7 (1997), no. 3, 403–419.Google Scholar
[5] Bonk, M. and Schramm, O. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10 (2000), no. 2, 266–306.CrossRefGoogle Scholar
[6] Bourdon, M. and Pajot, H. Cohomologie ℓ p et espace de Besov. J. Reine Angew. Math. 558 (2003), 85–108.Google Scholar
[7] Bridson, M.R. and Haefliger, A. Metric Spaces of Non-Positive Curvature (Springer-Verlag, 1999).Google Scholar
[8] Buyalo, S., Dranishnikov, A. and Schroeder, V. Embedding of hyperbolic groups into products of binary trees. Invent. Math. 169 (2007), no. 1, 153–192.Google Scholar
[9] Buyalo, S. and Schroeder, V. Elements of Asymptotic Geometry. EMS Monogr. Math. (Zürich, 2007).Google Scholar
[10] Choucroun, F. Arbres, espaces ultramétriques et bases de structure uniforme. Geom. Dedicata 53 (1994), no. 1, 69–74.Google Scholar
[11] Coornaert, M., Delzant, T. and Papadopoulos, A. Notes sur les groupes hyperboliques de Gromov. Lecture Notes in Math., vol. 1441 (Springer-Verlag, 1990).Google Scholar
[12] Coornaert, M. and Papadopoulos, A. Symbolic dynamics and hyperbolic groups, Springer Lecture Notes, vol. 1539, Springer–Verlag, 1993.Google Scholar
[13] Elek, G. The ℓ p -cohomology and the conformal dimension of hyperbolic cones. Geom. Dedicata 68 (1997), no. 3, 263–279.Google Scholar
[14] Ghys, E. and Harpe, P. de la Sur les groupes hyperboliques, d'après M. Gromov. Progr. Math., vol. 83 (Birkhäuser, Boston, 1990).Google Scholar
[15] Gromov, M. Hyperbolic Groups. Essays in group theory (Gersten, S.M., ed.), MSRI, vol. 8 (Springer, New York, 1987), pp. 75–263).Google Scholar
[16] Gromov, M. Asymptotic invariants of infinite groups. London Math. Soc. Lecture Notes, vol. 182 (Cambridge University Press, 1993).Google Scholar
[17] Hamann, M. Spanning trees in hyperbolic graphs. Combinatorica 36 (2016), no. 3, 313–332.Google Scholar
[18] Hughes, B. Trees and ultrametric spaces: a categorical equivalence. Adv. Math. 189 (2004), no. 1, 148–191.Google Scholar
[19] Hughes, B. Trees, ultrametrics, and noncommutative geometry. Pure Appl. Math. Q. 8 (2012), no. 1, 221–312.Google Scholar
[20] Kapovich, I. and Benakli, N. Boundaries of hyperbolic groups. Combinatorial and Geometric Group Theory (Gilman, R. et al., ed.) Contemp. Math., vol. 296 (2002), pp. 39–94.Google Scholar
[21] Lang, U. and Schlichenmaier, T. Nagata dimension, quasisymmetric embeddings and Lipschitz extensions. Int. Math. Res. Not. 58 (2005), 3625–3655.Google Scholar
[22] Luukainen, J. Assouad Dimension: antifractal metrization, Porous sets, and homogeneous measures. J. Korean Math. Soc. 35 (1998), no. 1, 23–76.Google Scholar
[23] Ohshika, K. Discrete groups. Translated from the 1998 Japanese original by the author. Trans. Math. Monogr., vol. 207 Iwanami Series in Modern Mathematics (Amer. Math. Soc., Providence, RI, 2002).Google Scholar
[24] Paulin, F. Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94 (1988), no. 1, 53–80.Google Scholar
[25] Woess, W. Random Walks on Infinite Graphs and Groups (Cambridge University Press, 2000).Google Scholar