  We define a new class of irreducible groups, called groups not infinite-index
presentable by products or not IIPP. We prove that certain aspherical manifolds
with fundamental groups not IIPP do not admit maps of non-zero degree from
direct products. This extends previous results of Kotschick and Loeh, providing
new classes of aspherical manifolds - beyond those non-positively curved ones
which were predicted by Gromov - that do not admit maps of non-zero degree from
direct products.
  A sample application is that an aspherical geometric 4-manifold admits a map
of non-zero degree from a direct product if and only if it is a virtual product
itself. This completes a characterization of the product geometries due to
Hillman. Along the way we prove that for certain groups the property IIPP is a
criterion for reducibility. This especially implies the vanishing of the
simplicial volume of the corresponding aspherical manifolds. It is shown that
aspherical manifolds with reducible fundamental groups do always admit maps of
non-zero degree from direct products.
