Kleinian groups, Laplacian on forms and currents at infinity

Pollicott, Mark

doi:10.1090/S0002-9939-1990-1012936-5

Kleinian groups, Laplacian on forms and currents at infinity
HTML articles powered by AMS MathViewer

by Mark Pollicott PDF

Proc. Amer. Math. Soc. 110 (1990), 269-279 Request permission

Abstract:

In this note we consider the spectrum of the Laplacian acting on the space of (co-closed) differential forms on the quotient of $n$-dimensional hyperbolic space by a co-compact Kleinian group. Using a result of P.-Y. Gaillard we relate these to currents on the sphere at infinity of hyperbolic space with distinctive transformation properties under the action of the group. We analyse these currents using zeta-functions and Ruelle’s Transfer operator. This represents a partial extension of earlier work of the author related to Fuchsian groups. In an appendix we propose an alternative approach to related questions.

References

Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
Robert Brooks, On the deformation theory of classical Schottky groups, Duke Math. J. 52 (1985), no. 4, 1009–1024. MR 816397, DOI 10.1215/S0012-7094-85-05253-6
Robert Brooks, Injectivity radius and low eigenvalues of hyperbolic manifolds, J. Reine Angew. Math. 390 (1988), 117–129. MR 953681, DOI 10.1515/crll.1988.390.117
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Peter G. Doyle, On the bass note of a Schottky group, Acta Math. 160 (1988), no. 3-4, 249–284. MR 945013, DOI 10.1007/BF02392277
David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 491–517. MR 875085
Pierre-Yves Gaillard, Transformation de Poisson de formes differentielles. Le cas de l’espace hyperbolique, Comment. Math. Helv. 61 (1986), no. 4, 581–616 (French). MR 870708, DOI 10.1007/BF02621934
Ramesh Gangolli, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. Math. 21 (1977), no. 1, 1–41. MR 485702
Laurent Guillopé, Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J. 53 (1986), no. 3, 827–848 (French). MR 860674, DOI 10.1215/S0012-7094-86-05345-7
Sigurdur Helgason, Topics in harmonic analysis on homogeneous spaces, Progress in Mathematics, vol. 13, Birkhäuser, Boston, Mass., 1981. MR 632696
S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
R. S. Phillips and P. Sarnak, The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta Math. 155 (1985), no. 3-4, 173–241. MR 806414, DOI 10.1007/BF02392542

Some applications of thermodynamic formalism to compact manifolds of constant negative curvature

David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231–242. MR 420720, DOI 10.1007/BF01403069
Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586

Meromorphic continuations of Ruelle zeta functions

Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297