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

TL;DR

This paper demonstrates that the Word Problem for finitely generated subgroups of integer matrices can be solved efficiently on average, with linear complexity, under realistic computational models and random input assumptions.

Contribution

It establishes the first linear average-case complexity result for the Word Problem in matrix groups over integers and certain rings, extending previous worst-case analyses.

Findings

Word Problem solvable in linear average-case time

Results hold under realistic bit-complexity models

Applicable to matrices over rings with finite rank over integers

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 $\mathbb{Z}$.