  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 $M$, and
any measurable subset $\omega\subset M$ such that $M\backslash \omega$ contains
in its interior a point $q$ with $[X_i,X_j](q)\notin
\text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we show that for any
$T_0>0$, 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 geodesics (for the
associated sub-Riemannian distance) spending a long time in $M\backslash
\omega$. 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.
