×

On the Patterson-Sullivan measure of some discrete group of isometries. (English) Zbl 1017.37022

The author constructs a class of groups of isometries of the \(d\)-dimensional hyperbolic space which are not geometrically finite but have finite Patterson-Sullivan measure, thus answering a question raised by Sullivan. The same result was obtained by A. Ancona using methods based on potential theory. The groups constructed by the author are free products of two Kleinian groups which are in Schottky position, that is satisfying the hypothesis of the ping-pong lemma.

MSC:

37F40 Geometric limits in holomorphic dynamics
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Ancona,Exemples de surfaces hyperboliques de type divergent, de mesure de Sullivan associées finies mais non géométriquement finies, Preprint.
[2] Bers, L., On boundaries of Teichmüller spaces and on Kleinian groups, Annals of Mathematics, 91, 570-600 (1970) · Zbl 0197.06001 · doi:10.2307/1970638
[3] Bishop, C. J.; Jones, P. W., Hausdorff dimension and kleinian groups, Acta Mathematica, 179, 1-39 (1997) · Zbl 0921.30032 · doi:10.1007/BF02392718
[4] Bowditch, B., Geometrical finiteness with variable negative curvature, Duke Mathematical Journal, 77, 229-274 (1995) · Zbl 0877.57018 · doi:10.1215/S0012-7094-95-07709-6
[5] Dal’bo, F.; Otal, J. P.; Peigné, M., Séries de Poincaré des groupes géométriquement finis, Israel Journal of Mathematics, 118, 109-124 (2000) · Zbl 0968.53023
[6] Furusawa, H., Poincaré series of combination groups, The Tôhoku Mathematical Journal, 43, 1-7 (1991) · Zbl 0739.20007
[7] Hamilton, E., Geometrical finiteness for hyperbolic orbifolds, Topology, 37, 635-657 (1998) · Zbl 0915.32006 · doi:10.1016/S0040-9383(97)00043-8
[8] Maskit, B., Kleinian Groups (1987), Berlin: Springer-Verlag, Berlin
[9] Patterson, S. J., The limit set of a Fuchsian group, Acta Mathematica, 136, 241-273 (1976) · Zbl 0336.30005 · doi:10.1007/BF02392046
[10] Patterson, S. J., The exponent of convergence of Poincaré series, Monatshefte für Mathematik, 82, 297-315 (1976) · Zbl 0349.30012 · doi:10.1007/BF01540601
[11] Th. Roblin,Sur la théorie ergodique des groupes discrets en géométrie hyperbolique, Thèse de doctorat de l’Université d’Orsay, 1999.
[12] Sullivan, D., The density at infinity of a discrete group of hyperbolic motions, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 50, 171-202 (1979) · Zbl 0439.30034 · doi:10.1007/BF02684773
[13] Sullivan, D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Mathematica, 153, 259-277 (1984) · Zbl 0566.58022 · doi:10.1007/BF02392379
[14] Sullivan, D., Discrete conformal groups and measurable dynamics, Bulletin of the American Mathematical Society, 6, 57-73 (1984) · Zbl 0489.58027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.