No iterated identities satisfied by all finite groups

TL;DR

This paper proves that no non-trivial iterated identity is satisfied by all finite groups, providing bounds on the size of groups that do not satisfy a given identity, using polynomial dynamics methods.

Contribution

It establishes the non-existence of a universal iterated identity for finite groups and provides explicit bounds on counterexamples.

Findings

No iterated identity holds for all finite groups.

For any non-trivial word, a finite group of size at most exponential in the word length does not satisfy it.

Uses polynomial dynamics to construct counterexamples.

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 $\exp(l^C)$ 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.