  We examine the issue of renormalizability of asymptotically free field
theories on non-commutative spaces. As an example, we solve the non-commutative
O(N) invariant Gross-Neveu model at large N. On commutative space this is a
renormalizable model with non-trivial interactions. On the noncommutative
space, if we take the translation invariant ground state, we find that the
model is non-renormalizable. Removing the ultraviolet cutoff yields a trivial
non-interacting theory.
