Abstract
In this note, we employ infinite ergodic theory to derive estimates for the algebraic growth rate of the Poincaré series for a Kleinian group at its critical exponent of convergence.
Dedicated to S.J. Patterson on the occasion of his 60th birthday.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
J. Aaronson. An introduction to infinite ergodic theory. Mathematical Surveys and Monographs 50, American Mathematical Society, 1997.
J. Aaronson, M. Denker, M. Urbański. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (2): 495–548, 1993.
A. F. Beardon. The exponent of convergence of Poincaré series. Proc. London Math. Soc. (3) 18: 461–483, 1968.
R. Bowen, C. Series. Markov maps associated with Fuchsian groups. Publ. Math., Inst. Hautes Etud. Sci. 50:153–170, 1979.
M. Denker, B.O. Stratmann. Patterson measure: classics, variations and applications.
D. Epstein, C. Petronio. An exposition of Poincare’s polyhedron theorem. Enseign. Math., II. Ser. 40: 113–170, 1994.
J. Fiala, P. Kleban. Intervals between Farey fractions in the limit of infinite level. Ann. Sci. Math. Qubec 34 (1), 63–71, 2010.
M. Kesseböhmer, B.O. Stratmann. A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dynam. Sys., 24 (01):141–170, 2004.
M. Kesseböhmer, B.O. Stratmann. On the Lebesgue measure of sum-level sets for continued fractions. To appear in Discrete Contin. Dyn. Syst.
M. Kesseböhmer, B.O. Stratmann. A dichotomy between uniform distributions of the Stern-Brocot and the Farey sequence. To appear in Unif. Distrib. Theory. 7 (2), 2011.
P. Nicholls. The ergodic theory of discrete groups, London Math. Soc. Lecture Note Series 143, Cambr. Univ. Press, Cambridge, 1989.
S.J. Patterson. The limit set of a Fuchsian group. Acta Math. 136 (3–4): 241–273, 1976.
C. Series. The modular surface and continued fractions. J. London Math. Soc. (2) 31 (1): 69–80, 1985.
C. Series. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergod. Th. & Dynam. Sys. 6: 601–625, 1986.
M. Stadlbauer. The Bowen–Series map for some free groups. Dissertation 2002, University of Göttingen. Mathematica Gottingensis 5: 1–53, 2002.
M. Stadlbauer. The return sequence of the Bowen–Series map associated to punctured surfaces. Fundamenta Math. 182: 221–240, 2004.
M. Stadlbauer, B.O. Stratmann. Infinite ergodic theory for Kleinian groups. Ergod. Th. & Dynam. Sys. 25: 1305–1323, 2005.
B.O. Stratmann, S.L. Velani. The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. London Math. Soc. (3) 71 (1): 197–220, 1995.
D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. Inst. Haut. Études Sci. Publ. Math. 50: 171–202, 1979.
D. Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153: 259–277, 1984.
D. Sullivan. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. 6: 57–73, 1982.
Acknowledgements
We would like to thank the Mathematische Institut der Universität Göttingen for the warm hospitality during our research visit in Summer 2009. In particular, we would like to thank P. Mihailescu and S.J. Patterson for the excellent organization of the International Conference: Patterson 60 ++, which took place during the period of our visit.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this paper
Cite this paper
Kesseböhmer, M., Stratmann, B.O. (2012). A Note on the Algebraic Growth Rate of Poincaré Series for Kleinian Groups. In: Blomer, V., Mihăilescu, P. (eds) Contributions in Analytic and Algebraic Number Theory. Springer Proceedings in Mathematics, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1219-9_10
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1219-9_10
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-1218-2
Online ISBN: 978-1-4614-1219-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)