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Meromorphic extension of the Selberg zeta function for Kleinian groups via thermodynamic formalism

Published online by Cambridge University Press:  24 October 2008

André Rocha
André Rocha
Affiliation:
Departamento de Matemática. CCEN, Universidade Federal de Pernambuco Av. Professor Luiz Freire s/n, CEP: 50670-901, Recife-PE, Brazil
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Abstract

We prove the existence of a piecewise analytic expanding map associated to certain Kleinian groups without parabolics acting in the 3-dimensional hyperbolic space. These groups have a fundamental domain ℛ with the property that the geodesic planes containing each face are part of the tesselation. We use this map together with the methods of thermodynamic formalism to give another proof that the Selberg zeta function for such groups has a meromorphic extension to ℂ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 179 - 190
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Ahlfors, L.. Möbius transformations in several dimensions. Ordway Professorship Lectures in Mathematics (University of Minnesota, 1981).Google Scholar
[2]Bourdon, M.. Actions quasi-convexes d'un groupe hyperbolique, flot géodésique (Ph.D. thesis, Université de Paris Sud, 1993).Google Scholar
[3]Bowen, R.. Hausdorff dimension of quasi-circles. I.H.E.S., Publ. Math. 50 (1979), 1–25.Google Scholar
[4]Floyd, W.. Group completions and limit sets of Kleinian groups. Inventiones Math. 57 (1980), 205–218.CrossRefGoogle Scholar
[5]Fried, D.. The zeta functions of Ruelle and Selberg I. Ann. Sci. E.N.S. 19 (1986), 491–517.Google Scholar
[6]Grothendieck, A.. La théorie de Fredhom. Bull. Soc. Math. France 84 (1956), 319–384.Google Scholar
[7]Lima, E.. Curso de Análise, vol. 2 (IMPA, 1981).Google Scholar
[8]Maskit, B.. Kleinian groups (Springer-Verlag, 1988).Google Scholar
[9]Mayer, D.. Continued fractions and related transformations; in Ergodic theory, symbolic, dynamics and hyperbolic geometry (ed. Keane, M., Bedford, T. and Series, C.), pp. 175–222 (Oxford University Press, 1991).Google Scholar
[10]Mayer, D.. The thermodynamic formalism approach to Selberg's zeta function for PSL(2, ℤ). Bull. Amer. Math. Soc. 25 (1991), 55–60.CrossRefGoogle Scholar
[11]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–88 (1990), 1–268.Google Scholar
[12]Pollicott, M.. Some applications of the thermodynamic formalism to manifolds of constant negative curvature. Adv. in Math. 85 (1991), 161–192.CrossRefGoogle Scholar
[13]Pollicott, M.. The Picard group, closed geodesies and zeta functions. Trans. Amer. Math. Soc. 344 (1994), 857–872.Google Scholar
[14]Ruelle, D.. Zeta-functions for expanding maps and Anosov flows. Inventiones. Math. 34 (1976), 231–242.CrossRefGoogle Scholar
[15]Series, C.. The infinite word problem and limit sets in Fuchsian groups. Ergod. Th. and Dynam. Sys. 1 (1981), 337–360.CrossRefGoogle Scholar
[16]Series, C.. Geometrical methods of symbolic dynamics; in Ergodic theory, symbolic dynamics and hyperbolic geometry (ed. Keane, M., Bedford, T. and Series, C.), pp. 125–151 (Oxford University Press, 1991).Google Scholar