Duke Mathematical Journal

Isospectrality and commensurability of arithmetic hyperbolic $2$- and $3$-manifolds

Alan W. Reid

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 65, Number 2 (1992), 215-228.

First available in Project Euclid: 20 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58G25
Secondary: 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 57N05: Topology of $E^2$ , 2-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]


Reid, Alan W. Isospectrality and commensurability of arithmetic hyperbolic 2 - and 3 -manifolds. Duke Math. J. 65 (1992), no. 2, 215--228. doi:10.1215/S0012-7094-92-06508-2. http://projecteuclid.org/euclid.dmj/1077295133.

Export citation


  • [AHW] C. Adams, M. Hildebrand, and J. Weeks, Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc. 326 (1991), no. 1, 1–56.
  • [AT] E. Artin and J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
  • [B] A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983.
  • [BB] L. Bérard-Bergery, Laplacien et géodésiques fermées sur les formes d'espace hyperbolique compactes, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 406, Springer, Berlin, 1973, 107–122. Lecture Notes in Math., Vol. 317.
  • [BZ] G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985.
  • [Br] R. Brooks, Constructing isospectral manifolds, Amer. Math. Monthly 95 (1988), no. 9, 823–839.
  • [Bu] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 167–192.
  • [C] S. Chen, Constructing isospectral but non-isometric Riemannian manifolds, preprint.
  • [Ga] R. Gangolli, The length spectra of some compact manifolds of negative curvature, J. Differential Geom. 12 (1977), no. 3, 403–424.
  • [G] R. Guralnick, Subgroups inducing the same permutation representation, J. Algebra 81 (1983), no. 2, 312–319.
  • [MR] C. Maclachlan and A. W. Reid, Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 251–257.
  • [M1] R. Meyerhoff, A lower bound for the volume of hyperbolic $3$-manifolds, Canad. J. Math. 39 (1987), no. 5, 1038–1056.
  • [M2] R. Meyerhoff, The ortho-length spectrum for hyperbolic $3$-manifolds, preprint.
  • [Mc] H. P. McKean, Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225–246.
  • [Mi] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542.
  • [NR] W. D. Neumann and A. W. Reid, Arithmetic of hyperbolic $3$-manifolds, to appear in Topology '90, Proceedings Of The Research Semester In Low Dimensional Topology At The Ohio State University.
  • [P] C. J. Parry, Units of algebraic numberfields, J. Number Theory 7 (1975), no. 4, 385–388.
  • [R] A. W. Reid Ph.D. thesis, Univ. of Aberdeen, 1987.
  • [Rib] P. Ribenboim, Algebraic numbers, Wiley-Interscience [A Division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972.
  • [Ri] R. Riley, Parabolic representations and symmetries of the knot $9\sb 32$, Computers in geometry and topology (Chicago, IL, 1986), Lecture Notes in Pure and Appl. Math., vol. 114, Dekker, New York, 1989, pp. 297–313.
  • [Sp]1 R. J. Spatzier, On isospectral locally symmetric spaces and a theorem of von Neumann, Duke Math. J. 59 (1989), no. 1, 289–294.
  • [Sp]2 R. J. Spatzier, Correction to: “On isospectral locally symmetric spaces and a theorem of von Neumann”, Duke Math. J. 60 (1990), no. 2, 561.
  • [S] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169–186.
  • [T1] K. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600–612.
  • [T2] K. Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 201–212.
  • [T] W. Thurston, The geometry and topology of $3$-manifolds, mimeographed lecture notes, Princeton Univ., 1978.
  • [V1] M.-F. Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112 (1980), no. 1, 21–32.
  • [V2] M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math., vol. 800, Springer, Berlin, 1980.
  • [Z] R. Zimmer, Ergodic theory and semisimple Lie groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984.