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Compact groups with a set of positive Haar measure satisfying a nilpotent law

Part of: Probabilistic methods in group theory Compact groups Abstract harmonic analysis

Published online by Cambridge University Press:  19 July 2021

ALIREZA ABDOLLAHI  and
MEISAM SOLEIMANI MALEKAN
ALIREZA ABDOLLAHI
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran. e-mails: a.abdollahi@math.ui.ac.ir, msmalekan@gmail.com
MEISAM SOLEIMANI MALEKAN
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran. e-mails: a.abdollahi@math.ui.ac.ir, msmalekan@gmail.com
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Abstract

The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]:

Let G be a compact group, and suppose that

\[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\]
has positive Haar measure in $G^{k+1}$. Does G have an open k-step nilpotent subgroup?

We give a positive answer for $k = 2$.

MSC classification

Primary: 22C05: Compact groups 43A05: Measures on groups and semigroups, etc.
Secondary: 20P05: Probabilistic methods in group theory

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 329 - 332
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

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