Stable complexity and simplicial volume of manifolds

Let the Delta-complexity sigma(M) of a closed manifold M be the minimal 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 sigma to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we denote by sigma(infinity)(M) and call the stable Delta-complexity of M. \ We study here the relation between the stable Delta-complexity sigma(infinity)(M) of M and Gromov's simplicial volume vertical bar vertical bar M vertical bar vertical bar. It is immediate to show that vertical bar vertical bar M vertical bar vertical bar < sigma(infinity) (M) and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental groups. We show that this is not always the case: there is a constant C-n < 1 such that vertical bar vertical bar M vertical bar vertical bar < C-n sigma(infinity)(M) for any hyperbolic manifold M of dimension n >= 4. The question in dimension 3 is still open in general. We prove that sigma(infinity)(M) = vertical bar vertical bar M vertical bar vertical bar 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.