  Compact (ferro- and antiferromagnetic) sigma-models and noncompact
(hyperbolic) sigma-models are compared in a lattice formulation in dimensions
$d \geq 2$. While the ferro- and antiferromagnetic models are essentially
equivalent, the qualitative difference to the noncompact models is highlighted.
The perturbative and the large $N$ expansions are studied in both types of
models and are argued to be asymptotic expansions on a finite lattice. An exact
correspondence between the expansion coefficients of the compact and the
noncompact models is established, for both expansions, valid to all orders on a
finite lattice. The perturbative one involves flipping the sign of the coupling
and remains valid in the termwise infinite volume limit. The large $N$
correspondence concerns the functional dependence on the free propagator and
holds directly only in finite volume.
