Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds. (English) Zbl 1169.53030

Let \(M\) be a closed, orientable Riemannian manifold of negative curvature. The rational length spectrum \(QL(M)\) of \(M\) is the set of all rational multiples of lengths of closed geodesics of \(M\). The commensurability class of \(M\) is the set of all manifolds \(M'\) for which \(M\) and \(M'\) have a common finite unramified cover. The main result is:
Theorem. If \(M\) is an arithmetic hyperbolic 3-manifold, then the rational length spectrum and the commensurability class of \(M\) determine one another.
This paper sharpens the result in [A. W. Reid, Duke Math. J. 65, No. 2, 215–228 (1992; Zbl 0776.58040)], where it was proved that the complex length spectrum of \(M\) determines its commensurability class.


53C22 Geodesics in global differential geometry
58J53 Isospectrality
11R42 Zeta functions and \(L\)-functions of number fields


Zbl 0776.58040
Full Text: DOI


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