Asymptotically free theories based on discrete subgroups

We study the critical behavior of discrete spin models related to the 2d O(3) non-linear sigma model. Precise numerical results suggest that models with sufficiently large discrete subgroups are in the same universality class as the original sigma model. We observe that at least up to correlation lengths $\xi\approx 300$ the cut-off effects follow effectively an $\propto a$ behaviour both in the O(3) and in the dodecahedron model.