Bounding the residual finiteness of free groups

Abstract

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 $\sim n^{2/3}$ or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.