Short Laws for Finite Groups and Residual Finiteness Growth

We prove that for every $n\in \mathbb{N}$ and $\delta >0$ there exists a word $w_n\in F_2$ of length $n^{2\mathrm{/}3}\log (n{)}^{3+\delta }$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom, About the length of laws for finite groups, Isr. J. Math.]. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups.