  The Krylov--Safonov theory for fully nonlinear nonlocal operators on
hyperbolic spaces of dimension three is established. Since the operators on
hyperbolic spaces exhibit qualitatively different behavior than those on
manifolds with nonnegative curvature, new scale functions are introduced which
take the effect of negative curvature into account.
  The regularity theory in this work provides unified regularity results for
fractional-order and second-order operators in the sense that the regularity
estimates stay uniform as the fractional-order approaches 2. In the unified
regularity theory, the asymptotic behavior of the normalizing constant for the
fractional Laplacian plays a fundamental role. The dimension restriction has
been imposed to compute the explicit value of this constant by using the
Fourier analysis on hyperbolic spaces.
