Direct numerical simulations of three-dimensional homogeneous turbulence under rapid rigid rotation are conducted for a fixed large reynolds number and a sequence of decreasing rossby numbers to examine the predictions of resonant wave theory. the theory states that ‘slow modes’ of the velocity, with zero wavenumber parallel to the rotation axis ($k_z{=}0$), will decouple at first order from the remaining ‘fast modes’ and solve an autonomous system of two-dimensional navier–stokes equations for the horizontal velocity components, normal to the rotation axis, and a two-dimensional passive scalar equation for the vertical velocity component, parallel to the rotation axis. The navier–stokes equation for three-dimensional rotating turbulence is solved in a $128^3$ mesh after being diagonalized via ‘helical decomposition’ into normal modes of the coriolis term. A force supplies constant energy input at intermediate scales. to verify the theory, we set up a corresponding simulation for the two-dimensional navier–stokes equation and two-dimensional passive scalar equation to compare them with the slow-mode dynamics of the three-dimensional rotating turbulence. the simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by two-dimensional navier–stokes equations for resonant interactions of slow modes. as the rotation rate increases, the vertically averaged horizontal velocity field from three-dimensional navier–stokes converges to the velocity field from two-dimensional navier–stokes, as measured by the energy in their difference field. likewise, the vertically averaged vertical velocity from three-dimensional navier–stokes converges to a solution of the two-dimensional passive scalar equation. the slow-mode energy spectrum approaches $k_h^{-5/3},$ where $k_h$ is the horizontal wavenumber, and, as in two dimensions, energy flux becomes closer to constant the greater the rotation rate. furthermore, the energy flux directly into small wavenumbers in the $k_z{=}0$ plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. the simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation.