Bounding the residual finiteness of free groups

Proceedings of the American Mathematical Society, 139(7) (2011), 2281--2286
http://dx.doi.org/10.1090/S0002-9939-2011-10967-5 (preprint - http://arxiv.org/abs/0912.2368)

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.