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arXiv · 2506.00948

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

Frédérique Bassino·Cyril Nicaud·Pascal Weil
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Abstract

We show that the Word Problem in finitely generated subgroups of $\textsf{GL}_d(\mathbb{Z})$ 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 $\textsf{GL}_d(R)$, where $R$ is a subring of $\mathbb{C}$, of finite rank over $\mathbb{Z}$.

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Frédérique Bassino, Cyril Nicaud, Pascal Weil. 2025-06-01. The average-case complexity of the Word Problem for groups of matrices over $\mathbb{Z}$ is linear. https://arxiv.org/abs/2506.00948

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