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Take a Riemannian nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the Dirichlet Laplacian. Using homogenization techniques we describe the asymptotic behavior of the spectrum when the radius of these balls goes to infinity. This involves the spectrum, which we call macroscopic spectrum, of a so called homogenized operator on a specific domain. Furthermore we show that the first macroscopic eigenvalue is bounded from above, by a universal constant in the case of the three dimensional Heisenberg group, and by a constant depending on the Albanese torus for the other nilmanifolds. We also show that the Heisenberg groups belong to a family of nilmanifolds, where the equality characterizes some pseudo-left-invariant metrics

The macroscopic spectrum of nilmanifolds with an emphasis on the Heisenberg groups