  We consider spaces of high-energy quasimodes for the Laplacian on a compact
hyperbolic surface, and show that when the spaces are large enough, one can
find quasimodes that exhibit strong localization phenomena. Namely, take any
constant c, and a sequence of (cr_j)-dimensional spaces S_j of quasimodes,
where 1/4+r_j^2 is an approximate eigenvalue for S_j. Then we can find a vector
psi_j in each S_j, such that any weak-* limit point of the microlocal lifts of
|psi_j|^2 localizes a positive proportion of its mass on a singular set of
codimension 1. This result is sharp, in light of recent joint work with E.
Lindenstrauss, proving QUE for certain joint quasimodes that include spaces of
size o(r_j) with arbitrarily slow decay.
