  This paper studies Liouville properties for viscosity sub- and supersolutions
of fully nonlinear degenerate elliptic PDEs, under the main assumption that the
operator has a family of generalized subunit vector fields that satisfy the
H\"ormander condition. A general set of sufficient conditions is given such
that all subsolutions bounded above are constant; it includes the existence of
a supersolution out of a big ball, that explodes at infinity. Therefore for a
large class of operators the problem is reduced to finding such a Lyapunov-like
function. This is done here for the vector fields that generate the Heisenberg
group, giving explicit conditions on the sign and size of the first and zero-th
order terms in the equation. The optimality of the conditions is shown via
several examples. A sequel of this paper applies the methods to other Carnot
groups and to Grushin geometries.
