Commensurability and virtual fibration for graph manifolds
نویسنده
چکیده
Two manifolds are commensurable if they have diffeomorphic covers. We would like invariants that distinguish manifolds up to commensurability. A collection of such commensurability invariants is complete if it always distinguishes non-commensurable manifolds. Commensurability invariants of hyperbolic 3-manifolds are discussed in [NRe]. The two main ones are the invariant trace field and the invariant quaternion algebra. The latter is a complete commensurability invariant in the arithmetic case, but not in general. The set of primes at which traces fail to be integral is another commensurability invariant, and examples are given in [NRe] where the invariant quaternion algebras agree but this set does not. Another commensurability invariant discussed in [NRe] of the collection of “cusp fields” (the fields generated by cusp parameters). Craig Hodgson has pointed out that the set of PSL(2;Q)-classes of a cusp parameters is a finer commensurability invariant than the cusp fields when the degree of some cusp field exceeds 3. Here we discuss commensurability of non-hyperbolic 3-manifolds. For 3-manifolds with geometric structure the classification is known (cf. sec. 1):
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