The study of sup norms of eigenfunctions of the Laplacian on compact manifolds has a long history, the first results dating back to the 60's and the work of Hormander. When the compact manifold is a negatively curved arithmetic locally symmetric space, sup norms of eigenfunctions have attracted the attention of number theorists, not least for their relation to L-functions. We shall be interested in the size of cusp forms on certain non-compact spaces, namely, congruence quotients of $SL(n)/SO(n)$. These eigenfunctions oscillate on a sizable bulk of the space and decay rapidly in the cusps. In transitioning between these two regimes, the oscillation slows and the form gets large. When $n=2$, Iwaniec and Sarnak quantified this behavior for Maass cusp forms, showing, in particular, that their sup norm grows as a power of the eigenvalue. In ongoing work with N. Templier, we investigate the size of cusp forms in the transition range in higher rank. Among other results, we obtain lower bounds on the sup norms of surprising strength for general $n$.