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

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

Duke Math. J. 145, No. 1, 25-44 (2008).

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.

Reviewer: Mikhail Malakhal’tsev (Kazan)

MSC:

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

Keywords:

arithmetic hyperbolic 3-manifold; rational length spectrum; commensurability class

Citations:

Zbl 0776.58040

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