  On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we
prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power
saving over the local geometric bound simultaneously in the Laplacian
eigenvalue and the volume. By a novel combination of diophantine and geometric
arguments in a noncommutative setting, we obtain bounds as strong as the best
corresponding results on arithmetic surfaces.
