Stable complexity and simplicial volume of manifolds

Abstract

Let the Δ-complexity σ(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 σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we denote by σ∞(M) and call the stable Δ-complexity of M. We study here the relation between the stable Δ-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 aspherical manifolds with residually finite fundamental groups. We show that this is not always the case: there is a constant Cn < 1 such that ||M|| ⩽ Cnσ∞(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.