Bradford, HenryHenryBradfordThom, AndreasAndreasThom2020-12-102020-12-1020190002-9947https://resolver.sub.uni-goettingen.de/purl?gro-2/75043We prove that for every n ∈ N n \in \mathbb {N} and δ > 0 \delta >0 there exists a word w n ∈ F 2 w_n \in F_2 of length O ( n 2 / 3 log ⁡ ( n ) 3 + δ ) O(n^{2/3} \log (n)^{3+\delta }) which is a law for every finite group of order at most n n . This improves upon the main result of Andreas Thom [Israel J. Math. 219 (2017), pp. 469–478] by the second named author. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups.enjournal_article10.1090/tran/75181088-6850S0002-9947-2018-07518-3