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The Patterson–Sullivan measure and proper conjugation for Kleinian groups of divergence type

Published online by Cambridge University Press:  01 April 2009

Department of Mathematics, Okayama University, Okayama 700-8530, Japan (email:
Mathematical Institute, Tohoku University, Aoba-ku, Sendai 980-8578, Japan (email:


A Kleinian group (a discrete subgroup of conformal automorphisms of the unit ball) G is said to have proper conjugation if it contains the conjugate αGα−1 by some conformal automorphism α as a proper subgroup in it. We show that a Kleinian group of divergence type cannot have proper conjugation. Uniqueness of the Patterson–Sullivan measure for such a Kleinian group is crucial to our proof.

Research Article
Copyright © Cambridge University Press 2008

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[1]Culler, M. and Shalen, P.. Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds. J. Amer. Math. Soc. 5 (1992), 231288.CrossRefGoogle Scholar
[2]Brooks, R.. The bottom of the spectrum of a Riemannian covering. J. Reine Angew. Math. 357 (1985), 101114.Google Scholar
[3]Fujikawa, E., Matsuzaki, K. and Taniguchi, M.. Dynamics on Teichmüller spaces and self-covering of Riemann surfaces. Math. Z. 260(4) (2008), to appear.CrossRefGoogle Scholar
[4]Heins, M.. On a problem of Heinz Hopf. J. Math. Pures Appl. 37 (1958), 153160.Google Scholar
[5]Jørgensen, T., Marden, A. and Pommerenke, C.. Two examples of covering surfaces. Riemann Surfaces and Related Topics (Annals of Mathematics Studies, 97). Princeton University Press, Princeton, 1978, pp. 305319.Google Scholar
[6]Matsuzaki, K.. Dynamics of Kleinian groups—the Hausdorff dimension of limit sets. Selected Papers on Classical Analysis (AMS Translation Series (2), 204). The American Mathematical Society, Providence, RI, 2001, pp. 2344.Google Scholar
[7]Matsuzaki, K. and Yabuki, Y.. Invariance of the Nayatani metrics for Kleinian manifolds. Geom. Dedicata 135 (2008), 147155.CrossRefGoogle Scholar
[8]McMullen, C. and Sullivan, D.. Quasiconformal homeomorphisms and dynamics III. The Teichmüller space of a holomorphic dynamical system. Adv. Math. 135 (1998), 351395.CrossRefGoogle Scholar
[9]Nayatani, S.. Patterson–Sullivan measure and conformally flat metrics. Math. Z. 225 (1997), 115131.CrossRefGoogle Scholar
[10]Nicholls, P.. The Ergodic Theory of Discrete Groups (LMS Lecture Note Series, 143). Cambridge University Press, Cambridge, 1989.CrossRefGoogle Scholar
[11]Ohshika, K. and Potyagailo, L.. Self-embeddings of Kleinian groups. Ann. Sci. Ecole Norm. Sup. 31 (1998), 329343.CrossRefGoogle Scholar
[12]Roblin, T.. Ergodicité et équidistribution en courbure négative (Mémoires de la Société Mathématique de France, 95). Société Mathématique de France, Paris, 2003.CrossRefGoogle Scholar
[13]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171202.CrossRefGoogle Scholar
[14]Sullivan, D.. Growth of Positive Harmonic Functions and Kleinian Group Limit Sets of Zero Planar Measure and Hausdorff Dimension Two (Geometry Symp., Utrecht, 1980) (Lecture Notes in Mathematics, 894). Springer, Berlin, 1981, pp. 127144.Google Scholar
[15]Wang, S. and Zhou, Q.. On the proper conjugation of Kleinian groups. Geom. Dedicata 56 (1995), 145154.CrossRefGoogle Scholar
[16]Wang, X. and Yang, W.. Discreteness criteria of Möbius groups of high dimensions and convergence theorems of Kleinian groups. Adv. Math. 159 (2001), 6882.CrossRefGoogle Scholar