  We study the statistical mechanics and dynamics of crystalline films with a
fixed internal connectivity on a random substrate. Defect free triangular
lattices exhibit a sharp transition to a low temperature glassy phase with
anomalous phonon fluctuations and a nonlinear force-displacement law with a
continuously variable exponent, similar to the vortex glass phase of directed
lines in 1+1 dimensions. The periodicity of the tethered monolayer acts like a
filter which amplifies particular Fourier components of the disorder. However,
the absence of annealed topological defects like dislocations is crucial: the
transition is destroyed when the constraint of fixed connectivity is relaxed
and dislocations are allowed to proliferate.
