Duke Mathematical Journal

Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds

T. Chinburg, E. Hamilton, D. D. Long, and A. W. Reid

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We show that if M is an arithmetic hyperbolic 3-manifold, the set QL(M) of all rational multiples of lengths of closed geodesics of M both determines and is determined by the commensurability class of M. This implies that the spectrum of the Laplace operator of M determines the commensurability class of M. We also show that the zeta function of a number field with exactly one complex place determines the isomorphism class of the number field

Article information

Duke Math. J., Volume 145, Number 1 (2008), 25-44.

First available in Project Euclid: 17 September 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C22: Geodesics [See also 58E10] 58J53: Isospectrality
Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]


Chinburg, T.; Hamilton, E.; Long, D. D.; Reid, A. W. Geodesics and commensurability classes of arithmetic hyperbolic $3$ -manifolds. Duke Math. J. 145 (2008), no. 1, 25--44. doi:10.1215/00127094-2008-045. https://projecteuclid.org/euclid.dmj/1221656861

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