Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

Abstract

On a family of arithmetic hyperbolic $3$-manifolds of square-free 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.