  We show that main-sequence stars in dense stellar cusps around massive black
holes are likely to rotate at a significant fraction of the centrifugal breakup
velocity due to spin-up by hyperbolic tidal encounters. We use realistic
stellar structure models to calculate analytically the tidal spin-up in soft
encounters, and extrapolate these results to close and penetrating collisions
using smoothed particle hydrodynamics simulations. We find that the spin-up
falls off only slowly with distance from the black hole because the increased
tidal coupling in slower collisions at larger distances compensates for the
decrease in the stellar density. We apply our results to the stars near the
massive black hole in the Galactic Center. Over their lifetime, ~1 Msol main
sequence stars in the inner 0.3 pc of the Galactic Center are spun-up on
average to ~10%--30% of the centrifugal breakup limit. Such rotation is ~20--60
times higher than is usual for such stars and may affect their subsequent
evolution and their observed properties.
