  We prove a new omega result for toric periods of Hecke-Maass forms on compact
locally symmetric spaces associated to forms of PGL(3). This is motivated by
conjectures on the maximal growth of L-functions as well as by questions about
the size of automorphic periods. We also prove a mean square asymptotic result
for maximal flat periods on more general locally symmetric spaces of
non-compact type, which takes as main input bounds for real relative orbital
integrals.
