SHORT LAWS FOR FINITE GROUPS AND RESIDUAL FINITENESS GROWTH.

We prove that for every n ∈ N and δ > 0 there exists a word wn ∈ F2 of length O(n2/3 log(n)3+δ) which is a law for every finite group of order at most 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. [ABSTRACT FROM AUTHOR]