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 $T_0$ almost equivalent to Geometric Control Condition (GCC), which stipulates any geodesic ray meets control set within $T_0$. We show subelliptic setting, GCC never verified, and equations are observable finite time. More precisely, given Laplacian $\Delta=-\sum_{i=1}^m X_i^*X_i$ on manifold $M$, measurable subset $\omega\subset M$ such $M\backslash \omega$ contains its interior point $q$ $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we $T_0>0$, equation $\Delta$ not $\omega$ The proof based construction sequences solutions concentrating geodesics (for associated sub-Riemannian distance) spending long \omega$. As counterpart, prove positive result Heisenberg group, where observation well-chosen part phase space.