  We show that a combination of collapsing and excessive growth from the
fundamental group impedes the existence of Einstein metrics on several families
of smooth four-manifolds. These include infrasolvmanifolds whose fundamental
group is not virtually nilpotent, most elliptic surfaces of zero Euler
characteristic, geometrizable manifolds with hyperbolic factor geometries in
their geometric decomposition, and higher graph four-manifolds without purely
negatively curved pieces.
