The Picard group, closed geodesics and zeta functions
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- by Mark Pollicott PDF
- Trans. Amer. Math. Soc. 344 (1994), 857-872 Request permission
Abstract:
In this article we consider the Picard group ${\text {SL}}(2,\mathbb {Z}[i])$, viewed as a discrete subgroup of the isometries of hyperbolic space. We fix a canonical choice of generators and then construct a Markov partition for the action of the group on the sphere at infinity. Our main application is to the study of the zeta function associated to the associated three-dimensional hyperbolic manifold.References
-
R. Adler and L. Flatto, Cross section maps for geodesic flows I, Ergodic Theory and Dynamical Systems II (A. Katok, ed.), Birkhäuser, Boston, MA, 1982.
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- Lester R. Ford, Rational approximations to irrational complex numbers, Trans. Amer. Math. Soc. 19 (1918), no. 1, 1–42. MR 1501085, DOI 10.1090/S0002-9947-1918-1501085-6
- Lester R. Ford, On the closeness of approach of complex rational fractions to a complex irrational number, Trans. Amer. Math. Soc. 27 (1925), no. 2, 146–154. MR 1501304, DOI 10.1090/S0002-9947-1925-1501304-X
- David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 491–517. MR 875085
- A. Hurwitz, Über die Entwicklung complexer Grössen in Kettenbrüche, Acta Math. 11 (1887), no. 1-4, 187–200 (German). MR 1554754, DOI 10.1007/BF02418048
- A. Grothendieck, La théorie de Fredholm, Bull. Soc. Math. France 84 (1956), 319–384 (French). MR 88665 J. Magnus, Non-eucildean tessilations and their groups, Academic Press, New York, 1974.
- Dieter H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for $\textrm {PSL}(2,\textbf {Z})$, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 55–60. MR 1080004, DOI 10.1090/S0273-0979-1991-16023-4
- Mark Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France 114 (1986), no. 4, 431–446. MR 882589
- David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231–242. MR 420720, DOI 10.1007/BF01403069
- P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 3-4, 253–295. MR 723012, DOI 10.1007/BF02393209
- Caroline Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69–80. MR 810563, DOI 10.1112/jlms/s2-31.1.69
- Asmus L. Schmidt, Farey triangles and Farey quadrangles in the complex plane, Math. Scand. 21 (1967), 241–295 (1969). MR 245525, DOI 10.7146/math.scand.a-10863
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 344 (1994), 857-872
- MSC: Primary 58F20; Secondary 11F06, 58F15, 58F17
- DOI: https://doi.org/10.1090/S0002-9947-1994-1240946-X
- MathSciNet review: 1240946