This question is related to this one . Let $G$ be a finitely generated subgroup of $GL_n(K)$ for some field $K$ of characteristic 0. Then $G$ is a linear group over $\mathbb{Q}(x_1,...,x_m)$, the field of rational functions over $\mathbb{Q}$. This follows from the fact that one can assume $K$ to be a finitely generated field, which is a finite extension of the field $\mathbb{Q}(x_1,...,x_m)$, and one can get rid of the finite extension by considering matrices of bigger size.

Question. What is the maximal possible computational complexity of the word problem in such $G$?

It seems clear that the complexity always is at most co-NP (one can check that a product of matrices with entries rational functions is not equal to 1 by plugging not very large values for the variables $x_1,...,x_m$ and computing the product of matrices over $\mathbb{Q}$. Is co-NP the best we can get in general (assuming $m\ge 2$, $n\ge 3$)?