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(lC) which does not satisfy the iterated identity w. We also prove a more general statement concerning iterations of an endomorphism of a free group. 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.
Bibliographical noteFunding Information:
The work of the authors is partially supported by the ERC grant GroIsRan. This work of the first-named author is also supported by the Russian Science Foundation grant No. 17-11-01377.
© 2019, The Hebrew University of Jerusalem.