Subelliptic wave equations are never observable

It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $Delta=-sum_{i=1}^m X_i^*X_i$ on a manifold $M$ such that $text{Lie}(X_1,ldots,X_m)=TM$ but $text{Span}(X_1,ldots,X_m)subsetneq TM$, we show that for any $T_0>0$ and any measurable subset $omegasubset M$ such that $Mbackslash omega$ has nonempty interior, the wave equation with subelliptic Laplacian $Delta$ is not observable on $omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on spiraling geodesics (for the associated sub-Riemannian distance) spending a long time in $Mbackslash omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.