Illinois Journal of Mathematics

On quasiconformal invariance of convergence and divergence types for Fuchsian groups

Katsuhiko Matsuzaki

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Abstract

We characterize convergence and divergence types for Fuchsian groups in terms of the critical exponent of convergence and modified functions of the Poincaré series for certain subgroups associated with ends of the quotient Riemann surfaces. As an application of this result, we prove that convergence and divergence type are not invariant under a quasiconformal automorphism of the unit disk.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1249-1258.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554360

Digital Object Identifier
doi:10.1215/ijm/1258554360

Mathematical Reviews number (MathSciNet)
MR2595765

Zentralblatt MATH identifier
1189.30081

Subjects
Primary: 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx] 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 37F35: Conformal densities and Hausdorff dimension

Citation

Matsuzaki, Katsuhiko. On quasiconformal invariance of convergence and divergence types for Fuchsian groups. Illinois J. Math. 52 (2008), no. 4, 1249--1258. doi:10.1215/ijm/1258554360. https://projecteuclid.org/euclid.ijm/1258554360


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References

  • L. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, 1960.
  • J. Anderson, K. Falk and P. Tukia, Conformal measures associated to ends of hyperbolic $n$-manifolds, Quart. J. Math. 58 (2007), 1–15.
  • C. Bishop and P. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), 1–39.
  • I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Orlando, FL, 1984.
  • K. Falk and B. Stratmann, Constructing restricted Patterson measures for geometrically infinite Kleinian groups, Acta Math. Sinica (Engl. Ser.) 22 (2006), 431–446.
  • K. Falk and P. Tukia, A note on Patterson measures, Kodai Math. J. 29 (2006), 227–236.
  • J. Fernández and J. Rodríguez, The exponent of convergence of Riemann surfaces. Bass Riemann surfaces, Ann. Acad. Sci. Fenn. 15 (1990), 165–183.
  • K. Matsuzaki, Ergodic properties of discrete groups; inheritance to normal subgroups and invariance under quasiconformal deformations, J. Math. Kyoto Univ. 33 (1993), 205–226.
  • K. Matsuzaki, Dynamics of Kleinian groups–-the Hausdorff dimension of limit sets, Selected papers on classical analysis, AMS Translation Series (2), vol. 204, Amer. Math. Soc., Providence, RI, 2001, pp. 23–44.
  • K. Matsuzaki, Isoperimetric constants for conservative Fuchsian groups, Kodai Math. J. 28 (2005), 292–300.
  • K. Matsuzaki and Y. Yabuki, The Patterson-Sullivan measure and proper conjugation for Kleinian groups of divergence type, Ergodic Theory Dynam. Systems 29 (2009), 657–665.
  • P. Nicholls, The ergodic theory of discrete groups, LMS Lecture Note Series, vol. 143, Cambridge Univ. Press, 1989.
  • S. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241–273.
  • L. Sario and M. Nakai, Classification theory of Riemann surfaces, Springer, New York, 1970.
  • B. Stratmann, The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones, Fractal geometry and stochastics III, Progr. Probab., vol. 57, Birkhäuser, Basel, 2004, pp. 93–107.
  • D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202.
  • D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), 327–351.
  • F. Dal'bo, J. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math. 118 (2000), 109–124.
  • K. Matsuzaki, A remark on the critical exponent of Kleinian groups, Analysis and geometry of hyperbolic spaces (Kyoto, 1997). Sūrikaisekikenkyūsho Kōkyūroku 1065 (1998), 106–107.