The Word Problem in finitely generated subgroups of $GL_d(Z)$ is the following: given a word on a fixed finite alphabet of invertible matrices with integer coefficients, decide whether this word evaluates to the identity matrix. The Word Problem is trivially decidable, in polynomial time. Its worst-case complexity is known to be $O(n log^2 n)$. We give an algorithm which solves it with linear average-case complexity. This is done under the bit-complexity model, which accounts for the fact that large integers are handled, and under the assumption that the input words are chosen uniformly at random among words of a given length.

This is joint work with Frédérique Bassino and Cyril Nicaud.