Cohomological induction and uniform measure equivalence

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, implies that the graded cohomology algebras of quasi-isometric, connected, simply connected nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low-dimensional nilpotent Lie groups.