  Pursuing the approach of Angel & Ray, we introduce and study a family of
random infinite triangulations of the full-plane that satisfy a natural spatial
Markov property. These new random lattices naturally generalize Angel &
Schramm's Uniform Infinite Planar Triangulation (UIPT) and are hyperbolic in
flavor. We prove that they exhibit a sharp exponential volume growth, are
non-Liouville, and that the simple random walk on them has positive speed
almost surely. We conjecture that these infinite triangulations are the local
limits of uniform triangulations whose genus is proportional to the size.
