  We investigate Gabor frames on locally compact abelian groups with
time-frequency shifts along non-separable, closed subgroups of the phase space.
Density theorems in Gabor analysis state necessary conditions for a Gabor
system to be a frame or a Riesz basis, formulated only in terms of the index
subgroup. In the classical results the subgroup is assumed to be discrete. We
prove density theorems for general closed subgroups of the phase space, where
the necessary conditions are given in terms of the "size" of the subgroup. From
these density results we are able to extend the classical Wexler-Raz
biorthogonal relations and the duality principle in Gabor analysis to Gabor
systems with time-frequency shifts along non-separable, closed subgroups of the
phase space. Even in the euclidean setting, our results are new.
