Commensurability and virtual fibration for graph manifolds

Abstract

Two manifolds are commensurable if they have diffeomorphic finite 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 [l]. 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 Cl] where the invariant quaternion algebras agree but this set does not. Another commensurability invariant discussed in [l] is 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 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 nonhyperbolic 3-manifolds. For 3-manifolds with geometric structure the classification is known (cf. Section 2): THEOREM A. For each of the six “Seifert geometries” S3, E3, S2 x IE’, E-U2 x [El, Nil, and PSL there is just one commensurability class of compact geometric 3-manifolds with the given structure (two for the last two geometries if orientation-preserving commensurability of oriented manifolds is considered). For the remaining nonhyperbolic geometry Sol, the commensurability classes are in one-one correspondence with real quadratic number jelds (such a manifold is covered by a torus bundle over a circle and the field in question is the field generated by an eigenvalue of the monodromy of this bundle). Noncompact finite volume nonhyperbolic geometric 3-manifolds admit geometric structures of both types W2 x lE3 and PSL and form just one commensurability class, also in the oriented case. In Section 3 we define several multiplicative invariants for prime 3-manifolds. A multiplicative invariant is one that multiplies by degree for covering spaces. Our invariants are most interesting for graph manifolds. Since the ratio of two multiplicative invariants is a commensurability invariant, we get several commensurability invariants also. The next two theorems, which use two of these invariants, are a start on the commensurability classification for manifolds with nontrivial geometric decomposition. Let M be a closed non-Seifert-fibered oriented 3-manifold obtained by pasting two Seifert manifolds, MI and M2, each having a torus as its boundary, along these tori. Suppose also that neither half Mi is the total space SMb of the circle bundle over the MSbius band with orientable total space (otherwise there is a double cover of M that either satisfies our requirements or is a torus bundle over the circle and is thus covered by Theorem A). To each