Bounding the residual finiteness of free groups

TL;DR

This paper establishes a new lower bound on the size of finite groups needed to detect certain words in free groups, improving previous quantifications of residual finiteness.

Contribution

It constructs specific words in free groups with controlled detection properties, advancing understanding of residual finiteness bounds.

Findings

Constructed words of length n with identity in all finite quotients of size up to n^{2/3}

Improved the lower bounds on residual finiteness of free groups

Quantified the detection limits of words in finite quotients

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 ~ 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.