This question is related to [this one](https://mathoverflow.net/questions/80336/the-word-problem-in-the-ring-of-polynomials) . 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 $m\ge 2$, $n\ge 3$)? 



The complexity of the word problem in linear groups
The word problem is in deterministic logspace for linear groups, so it is very fast! See Word problems solvable in logspace by Lipton and Zalcstein, J. Assoc. Comput. Mach. 24 (1977), no. 3, 522–526.


