The average-case complexity of the Word Problem for groups of matrices over $\mathbb{Z}$ is linear

We show that the Word Problem in finitely generated subgroups of ${\mathsf{\text{GL}}}_d(\mathbb{Z})$ can be solved in 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 the words of a given length. Our result generalizes to matrices in ${\mathsf{\text{GL}}}_d(R)$, where $R$ is a subring of $\mathbb{C}$, of finite rank over $\mathbb{Z}$.