Stable complexity and simplicial volume of manifolds

Let the ∆-complexity σ(M ) of a closed manifold M be the min- imal number of simplices in a triangulation of M . Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we denote by σ∞ (M ) and call the stable ∆- complexity of M . We study here the relation between the stable ∆-complexity σ∞ (M ) of M and Gromov’s simplicial volume M . It is immediate to show that M σ∞ (M ) and it is natural to ask whether the two quantities coincide on aspher- ical manifolds with residually finite fundamental group. We show that this is not always the case: there is a constant Cn < 1 such that M Cn σ∞ (M ) for any hyperbolic manifold M of dimension n 4. The question in dimension 3 is still open in general. We prove that σ∞ (M ) = M for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3- manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true.