  We study the random planar maps obtained from supercritical Galton--Watson
trees by adding the horizontal connections between successive vertices at each
level. These are the hyperbolic analog of the maps studied by Curien,
Hutchcroft and Nachmias in arXiv:1710.03137, and a natural model of random
hyperbolic geometry. We first establish metric hyperbolicity properties of
these maps: we show that they admit bi-infinite geodesics and satisfy a weak
version of Gromov-hyperbolicity. We also study the simple random walk on these
maps: we identify their Poisson boundary and, in the case where the underlying
tree has no leaf, we prove that the random walk has positive speed. Some of the
methods used here are robust, and allow us to obtain more general results about
planar maps containing a supercritical Galton--Watson tree.
