Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportional
ity principle for integral foliated simplicial volume for aspherical manifolds and give refined upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is complemented by the calculation of the integral foliated simplicial volume of Seifert 3-manifolds.
We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional
curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.
