Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Metric conformal structures and hyperbolic dimension


Author: Igor Mineyev
Journal: Conform. Geom. Dyn. 11 (2007), 137-163
MSC (2000): Primary 20F65, 20F67, 20F69, 37F35, 30C35, 54E35, 54E45, 51K99, 54F45
DOI: https://doi.org/10.1090/S1088-4173-07-00165-8
Published electronically: September 12, 2007
MathSciNet review: 2346214
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For any hyperbolic complex $ X$ and $ a\in X$ we construct a visual metric  $ \check{d}=\check{d}_a$ on $ \partial X$ that makes the $ \operatorname{Isom}(X)$-action on $ \partial X$ bi-Lipschitz, Möbius, symmetric and conformal.

We define a stereographic projection of  $ \check{d}_a$ and show that it is a metric conformally equivalent to  $ \check{d}_a$.

We also introduce a notion of hyperbolic dimension for hyperbolic spaces with group actions. Problems related to hyperbolic dimension are discussed.


References [Enhancements On Off] (What's this?)

  • 1. M. BONK AND B. KLEINER, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math., 150 (2002), pp. 127-183. MR 1930885 (2004k:53057)
  • 2. -, Rigidity for quasi-Möbius group actions, J. Differential Geom., 61 (2002), pp. 81-106.
  • 3. -, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geometry and Topology, 9 (2005), pp. 219-246.
  • 4. M. BOURDON, Structure conforme au bord et flot géodésique d'un $ {CAT}(-1)$-espace, Enseign. Math. (2), 41 (1995), pp. 63-102. MR 1341941 (96f:58120)
  • 5. M. R. BRIDSON AND A. HAEFLIGER, Metric spaces of non-positive curvature, vol. 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • 6. J. W. CANNON, The combinatorial riemann mapping theorem, Acta Math., 173 (1994), pp. 155-234. MR 1301392 (95k:30046)
  • 7. J. W. CANNON AND D. COOPER, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc., 330 (1992), pp. 419-431. MR 1036000 (92f:22017)
  • 8. J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY, Hyperbolic geometry, in Flavors of geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997, pp. 59-115. MR 1491098 (99c:57036)
  • 9. J. W. CANNON AND E. L. SWENSON, Recognizing constant curvature discrete groups in dimension $ 3$, Trans. Amer. Math. Soc., 350 (1998), pp. 809-849. MR 1458317 (98i:57023)
  • 10. M. COORNAERT, T. DELZANT, AND A. PAPADOPOULOS, Géométrie et théorie des groupes, vol. 1441 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1990.
    Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary. MR 1075994 (92f:57003)
  • 11. M. J. DUNWOODY, The accessibility of finitely presented groups, Invent. Math., 81 (1985), pp. 449-457. MR 807066 (87d:20037)
  • 12. W. J. FLOYD, Group completions and limit sets of Kleinian groups, Invent. Math., 57 (1980), pp. 205-218. MR 568933 (81e:57002)
  • 13. E. GHYS AND P. DE LA HARPE, Le bord d'un espace hyperbolique, in Sur les groupes hyperboliques d'après Mikhael Gromov, E. Ghys and P. de la Harpe, eds., vol. 83 of Progress in Mathematics, Birkhäuser, 1988, ch. 7.
  • 14. É. GHYS AND P. DE LA HARPE, eds., Sur les groupes hyperboliques d'après Mikhael Gromov, vol. 83 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1990.
    Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648 (92f:53050)
  • 15. M. GROMOV, Hyperbolic groups, in Essays in group theory, vol. 8 of Math. Sci. Res. Inst. Publ., Springer, New York, 1987, pp. 75-263. MR 919829 (89e:20070)
  • 16. M. GROMOV, Metric structures for Riemannian and non-Riemannian spaces, vol. 152 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1999.
    Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. MR 0682063 (85e:53051); MR 1699320 (2000d:53065)
  • 17. J. HEINONEN AND P. KOSKELA, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), pp. 1-61. MR 1654771 (99j:30025)
  • 18. S. HERSONSKY AND F. PAULIN, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv., 72 (1997), pp. 349-388. MR 1476054 (98h:58105)
  • 19. G. A. MARGULIS, The isometry of closed manifolds of constant negative curvature with the same fundamental group, Dokl. Akad. Nauk SSSR, 192 (1970), pp. 736-737. MR 0266103 (42:1012)
  • 20. I. MINEYEV, Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal. (GAFA), 11 (2001), pp. 807-839. MR 1866802 (2002k:20078)
  • 21. -, Flows and joins of metric spaces, Geometry and Topology, 9 (2005), pp. 402-482.
    Available at http://www.maths.warwick.ac.uk/gt/GTVol9/paper13.abs.html.
  • 22. I. MINEYEV AND G. YU, The Baum-Connes conjecture for hyperbolic groups, Invent. Math., 149 (2002), pp. 97-122. MR 1914618 (2003f:20072)
  • 23. J.-P. OTAL, Sur la géometrie symplectique de l'espace des géodésiques d'une variété à courbure négative, Rev. Mat. Iberoamericana, 8 (1992), pp. 441-456. MR 1202417 (94a:58077)
  • 24. P. PANSU, Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math., 14 (1989), pp. 177-212. MR 1024425 (90k:53079)
  • 25. P. SCOTT AND T. WALL, Topological methods in group theory, vol. 36 of LMS, 1979, pp. 137-203. MR 564422 (81m:57002)
  • 26. D. SULLIVAN, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), vol. 97 of Ann. of Math. Stud., Princeton, N.J., 1981, Princeton Univ. Press, pp. 465-496. MR 624833 (83f:58052)
  • 27. P. TUKIA, Differentiability and rigidity of Möbius groups, Invent. Math., 82 (1985), pp. 557-578. MR 811551 (87f:30058)
  • 28. -, On quasiconformal groups, J. Analyse Math., 46 (1986), pp. 318-346.
  • 29. P. TUKIA, A rigidity theorem for Möbius groups, Invent. Math., 97 (1989), pp. 405-431. MR 1001847 (90i:20051)
  • 30. P. TUKIA AND J. VÄISÄLÄ, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., 5 (1980), pp. 97-114. MR 595180 (82g:30038)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 20F65, 20F67, 20F69, 37F35, 30C35, 54E35, 54E45, 51K99, 54F45

Retrieve articles in all journals with MSC (2000): 20F65, 20F67, 20F69, 37F35, 30C35, 54E35, 54E45, 51K99, 54F45


Additional Information

Igor Mineyev
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Email: mineyev@math.uiuc.edu

DOI: https://doi.org/10.1090/S1088-4173-07-00165-8
Received by editor(s): May 7, 2007
Published electronically: September 12, 2007
Additional Notes: This project is partially supported by NSF CAREER grant DMS-0132514
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society