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June  2011, 31(2): 559-579. doi: 10.3934/dcds.2011.31.559

On some exotic Schottky groups

Marc Peigné 1,

1. 

Laboratoire de Mathématiques et Physique Théorique, Université Fran¸cois-Rabelais Tours, Fédération Denis Poisson - CNRS, Parc de Grandmont, 37200 Tours, France

Received  May 2010 Revised  February 2011 Published  June 2011

We construct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements that do not satisfy the so-called "parabolic gap condition'' introduced in [4]. This construction relies on the comparaison between the Poincaré series of these free groups and the potential of some transfer operator which appears naturally in this context.
Keywords: parabolic groups, Poincaré series., Schottky groups.
Mathematics Subject Classification: Primary: 37D40, 37C30; Secondary: 20H10, 53C2.
Citation: Marc Peigné. On some exotic Schottky groups. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 559-579. doi: 10.3934/dcds.2011.31.559
References:
[1]

M. Babillot and M. Peigné, Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps, Bull. Soc. Math. France, 134 (2006), 119-163.  Google Scholar

[2]

M. Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, Enseign. Math., 41 (1995), 63-102.  Google Scholar

[3]

K. Corlette and A. Iozzi, Limit sets of discrete groups of isometries of exotic hyperbolic spaces, Trans. Amer. Math. Soc., 351 (1999), 1507-1530. doi: 10.1090/S0002-9947-99-02113-3.  Google Scholar

[4]

F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel Jour. Math., 118 (2000), 109-124.  Google Scholar

[5]

F. Dal'bo, M. Peigné, J. C. Picaud and A. Sambusetti, On the growth of non-uniform lattices in pinched negatively curved manifolds, J. Reine Angew. Math., 627 (2009), 31-52.  Google Scholar

[6]

P. Eberlein, "Geometry of Non Positively Curved Manifolds,", Chicago Lectures in Mathematics., ().   Google Scholar

[7]

E. Ghys and P. de la Harpe, "Sur les groupes hyperboliques d'aprés Mickael Gromov (Bern 1988)" Progress Math., 83, Birkhäuser Boston, Boston, MA, 1990.  Google Scholar

[8]

V. Kaïmanovitch, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. IHP., 53 (1900), 361-393.  Google Scholar

[9]

J. P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. Journal, 125 (2004), 15-44. doi: 10.1215/S0012-7094-04-12512-6.  Google Scholar

[10]

M. Peigné, On the Patterson-Sullivan measure of some discrete group of isometries, Israel J. Math., 133 (2003), 77-88. doi: 10.1007/BF02773062.  Google Scholar

[11]

J. G. Ratcliffe, "Foundations of Hyperbolic Manifolds," 2nd Edition, Graduate Texts in Math. 149, Springer-Verlag, New York, 2006, 779 pages.  Google Scholar

[12]

Th. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003).  Google Scholar

[13]

B. Schapira, "Propriétés ergodiques du feuilletage horosphérique d'une variété à courbure négative," Ph.D thesis, Université d'Orléans, 2003, http://www.univ-orleans.fr/mapmo/publications/theses/SCHAPIRATHE.ps. Google Scholar

[14]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.  Google Scholar

[15]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379.  Google Scholar

[16]

C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005. doi: 10.1090/S0002-9947-96-01614-5.  Google Scholar

show all references

References:
[1]

M. Babillot and M. Peigné, Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps, Bull. Soc. Math. France, 134 (2006), 119-163.  Google Scholar

[2]

M. Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, Enseign. Math., 41 (1995), 63-102.  Google Scholar

[3]

K. Corlette and A. Iozzi, Limit sets of discrete groups of isometries of exotic hyperbolic spaces, Trans. Amer. Math. Soc., 351 (1999), 1507-1530. doi: 10.1090/S0002-9947-99-02113-3.  Google Scholar

[4]

F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel Jour. Math., 118 (2000), 109-124.  Google Scholar

[5]

F. Dal'bo, M. Peigné, J. C. Picaud and A. Sambusetti, On the growth of non-uniform lattices in pinched negatively curved manifolds, J. Reine Angew. Math., 627 (2009), 31-52.  Google Scholar

[6]

P. Eberlein, "Geometry of Non Positively Curved Manifolds,", Chicago Lectures in Mathematics., ().   Google Scholar

[7]

E. Ghys and P. de la Harpe, "Sur les groupes hyperboliques d'aprés Mickael Gromov (Bern 1988)" Progress Math., 83, Birkhäuser Boston, Boston, MA, 1990.  Google Scholar

[8]

V. Kaïmanovitch, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. IHP., 53 (1900), 361-393.  Google Scholar

[9]

J. P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. Journal, 125 (2004), 15-44. doi: 10.1215/S0012-7094-04-12512-6.  Google Scholar

[10]

M. Peigné, On the Patterson-Sullivan measure of some discrete group of isometries, Israel J. Math., 133 (2003), 77-88. doi: 10.1007/BF02773062.  Google Scholar

[11]

J. G. Ratcliffe, "Foundations of Hyperbolic Manifolds," 2nd Edition, Graduate Texts in Math. 149, Springer-Verlag, New York, 2006, 779 pages.  Google Scholar

[12]

Th. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003).  Google Scholar

[13]

B. Schapira, "Propriétés ergodiques du feuilletage horosphérique d'une variété à courbure négative," Ph.D thesis, Université d'Orléans, 2003, http://www.univ-orleans.fr/mapmo/publications/theses/SCHAPIRATHE.ps. Google Scholar

[14]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.  Google Scholar

[15]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379.  Google Scholar

[16]

C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005. doi: 10.1090/S0002-9947-96-01614-5.  Google Scholar

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