  We prove global well-posedness, scattering and blow-up results for
energy-subcritical focusing nonlinear Schr\"odinger equations on the hyperbolic
space. We show in particular the existence of a critical element for scattering
for all energy-subcritical power nonlinearities. For mass-supercritical
nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of
the equation in low dimension, below natural mass and energy thresholds given
by the ground states of the equation. The proofs are based on trapping by mass
and energy, compactness and rigidity, and are similar to the ones on the
Euclidean space, with a new argument, based on generalized Pohozaev identities,
to obtain appropriate monotonicity formulas.
