Preservation of higher cohomology operations by the averaged pullback isomorphism

Establish whether the cohomology algebra isomorphism \overline{\psi^*}: H^*(\mathfrak{h}) \to H^*(\mathfrak{g}) induced by an ergodic quasi-isometry \psi: G \to H preserves higher cohomology operations, for example by extending to a C_\infty–morphism. Confirming this would imply that the minimal models of H^*(\mathfrak{h}) and H^*(\mathfrak{g}) are isomorphic and, in turn, that the Lie algebras \mathfrak{g} and \mathfrak{h} are isomorphic.

Background

The authors construct an isomorphism of cohomology algebras using an ergodic limit of Lipschitz maps and amenable averages, recovering and generalizing known cohomological invariants under quasi-isometry. They note that invariance of higher operations beyond the cup product—encoded by C_\infty-structures—would be strong enough to identify minimal models and thereby fully classify nilpotent Lie groups up to isomorphism under quasi-isometry. Whether their isomorphism preserves such higher operations is not currently known.