  The purpose of this paper is to investigate the ground-state properties of
two-dimensional Heisenberg models on a square lattice with a given
dimerization. Our aim is threefold: First, we want to investigate the
dimensional transition from two to one dimension for three models consisting of
weakly coupled chains for large dimerizations. Simple scaling arguments show
that the interchain coupling is always relevant. The ground states of two of
these models therefore have one-dimensional nature only at the decoupling
point. The third considered model is more complicated, because it contains
additional relevant intrachain couplings leading to a gap as shown by scaling
arguments and numerical investigations. Second, we investigate at which point
the dimerization destroys the N\'eel ordered ground state of the isotropic
model. Within a mapping to a nonlinear sigma-model and linear spinwave theory
(LSWT) we conclude that the stability of the N\'eel ordered state depends on
the microscopic details of the model. Third, the considered models also can be
regarded as effective models for a spin system with spin-phonon coupling. This
leads to the question if a spin-Peierls transition, i.e. a gain of total energy
due to lattice distortion, is possible. LSWT shows that such a transition is
possible under certain conditions leading to a coexistence of long-range order
and spin-Peierls dimerization. We also find that the gain of magnetic energy is
largest for a stair-like distortion of the lattice.
