  Take a riemanniann 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
laplacian for the dirichlet problem on them. We describe the asymptotic
behaviour of the spectrum when the radius of these balls goes to infinity.
Furthermore we show that the first macroscopic eigenvalue is bounded from
above, by an uniform constant for the three dimensional heisenberg group, and
by a constant depending on the Albanese's 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.
