  The paper treats second order fully nonlinear degenerate elliptic equations
having a family of subunit vector fields satisfying a full-rank bracket
condition. It studies Liouville properties for viscosity sub- and
supersolutions in the whole space, namely, that under a suitable bound at
infinity from above and, respectively, from below, they must be constants. In a
previous paper we proved an abstract result and discussed operators on the
Heisenberg group. Here we consider various families of vector fields: the
generators of a Carnot group, with more precise results for those of step 2, in
particular H-type groups and free Carnot groups, the Grushin and the
Heisenberg-Greiner vector fields. All these cases are relevant in
sub-Riemannian geometry and have in common the existence of a homogeneous norm
that we use for building Lyapunov-like functions for each operator. We give
explicit sufficient conditions on the size and sign of the first and zero-th
order terms in the equations and discuss their optimality. We also outline some
applications of such results to the problem of ergodicity of multidimensional
degenerate diffusion processes in the whole space.
