Running couplings and triviality of field theories on noncommutative spaces

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