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Bounding the residual finiteness of free groups

Kassabov, Martin; Matucci, Francesco

Proceedings of the American Mathematical Society, 139(7) (2011), 2281--2286 (preprint -

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of size ~ n^{2/3} or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.