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No iterated identities satisfied by all finite groups

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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(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.

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, Ramat-Gan, 5290002, Israel

    Alexei Belov

  2. Département de athématiques et applications, École normale supérieure CNRS, PSL Research University, 45 rue d’Ulm, 75005, Paris, France

    Anna Erschler

Authors
  1. Alexei Belov
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  2. Anna Erschler
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Correspondence to Anna Erschler.

Additional 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.

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Belov, A., Erschler, A. No iterated identities satisfied by all finite groups. Isr. J. Math. 233, 167–197 (2019). https://doi.org/10.1007/s11856-019-1904-4

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  • DOI: https://doi.org/10.1007/s11856-019-1904-4

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