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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Short laws for finite groups and residual finiteness growth


Authors: Henry Bradford and Andreas Thom
Journal: Trans. Amer. Math. Soc. 371 (2019), 6447-6462
MSC (2010): Primary 20D06; Secondary 20E10
DOI: https://doi.org/10.1090/tran/7518
Published electronically: October 2, 2018
MathSciNet review: 3937332
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Abstract: 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.


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Additional Information

Henry Bradford
Affiliation: Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstra${\ss }$e 3-5, D-37073 Göttingen, Germany
MR Author ID: 1179863
Email: henry.bradford@mathematik.uni-goettingen.de

Andreas Thom
Affiliation: Institut für Geometrie, TU Dresden, D-01062 Dresden, Germany
MR Author ID: 780176
ORCID: 0000-0002-7245-2861
Email: andreas.thom@tu-dresden.de

Received by editor(s): March 10, 2017
Received by editor(s) in revised form: June 1, 2017, and December 4, 2017
Published electronically: October 2, 2018
Additional Notes: This research was supported by ERC Starting Grant No. 277728 and ERC Consolidator Grant Nos. 648329 and 681207.
Article copyright: © Copyright 2018 American Mathematical Society