Short laws for finite groups and residual finiteness growth

We prove that for every n \in \mathbb{N} and \delta >0 there exists a word w_n \in F_2 of length O(n^{2/3} \log (n)^{3+\delta }) 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 nam...

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Published in:Transactions of the American Mathematical Society 2019-05, Vol.371 (9), p.6447-6462
Main Authors: BRADFORD, HENRY, THOM, ANDREAS
Format: Article
Language:English
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Summary:We prove that for every n \in \mathbb{N} and \delta >0 there exists a word w_n \in F_2 of length O(n^{2/3} \log (n)^{3+\delta }) 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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7518