No iterated identities satisfied by all finite groups

Abstract

We show that there is no iterated identity satisfied by all finite groups. For w being a non-trivial word of length l, we show that there exists a finite group G of cardinality at most exppl C q which does not satisfy the iterated identity w. The proof uses the approach of Borisov and Sapir, who used dynamics of polynomial mappings for the proof of non residual finiteness of some groups.