Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds

This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas recalled in §2.1 and §2.2, each element of QL(M) ∪ QL(M ′) is a rational multiple of the logarithm of a real algebraic number. As noted by Prasad and Rapinchuk in [9], the Gelfond Schneider Theorem [1] implies that a ratio of such logarithms is transcendental if it is irrational. Thus if ` ∈ QL(M)−QL(M ′) then `/`′ is transcendental for all non-zero `′ ∈ QL(M ′). Recently Prasad and Rapinchuk have shown in [9] that if M is an arithmetic hyperbolic manifold of even dimension, then QL(M) and the commensurability class of M determine one another. In addition, they have shown that this is not always true for arithmetic hyperbolic 5-manifolds. However, they have announced a proof that for all locally symmetric spaces associated to a specified absolutely simple Lie group, there are only finitely many commensurability classes of arithmetic lattices giving rise to a given rational length spectrum. It is known (see [4] pp. 415–417) that for closed hyperbolic manifolds, the spectrum of the Laplace-Beltrami operator action on L2(M), counting multiplicities, determines the set of lengths of closed geodesics on M (without counting multiplicities). Hence Theorem 1.1 implies: