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 4-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 4-manifolds without purely negatively curved pieces.

PSS thankfully acknowledges partial support by DGAPA-UNAM PAPIIT grant IN104819.