A gap theorem for the ZL-amenability constant of a finite group

Document Type : Research Paper

Author

Lancaster University

Abstract

It was shown in [‎A‎. ‎Azimifard‎, ‎E‎. ‎Samei and N‎. ‎Spronk‎, ‎Amenability properties of the centres of group algebras‎, J‎. ‎Funct‎. ‎Anal.‎, ‎256 no‎. ‎5 (2009) 1544-1564‎.] that the ZL-amenability constant of a finite group is always at least $1$‎, ‎with equality if and only if the group is abelian‎. ‎It was also shown that for any finite non-abelian group this invariant is at least $301/300$‎, ‎but the proof relies crucially on a deep result of D‎. ‎A‎. ‎Rider on norms of central idempotents in group algebras‎.
 ‎Here we show that if $G$ is finite and non-abelian then its ZL-amenability constant is at least $7/4$‎, ‎which is known to be best possible‎. ‎We avoid use of Rider's reslt‎, ‎by analyzing the cases where $G$ is just non-abelian‎, ‎using calculations from [‎M‎. ‎Alaghmandan‎, ‎Y‎. ‎Choi and E‎. ‎Samei‎, ‎ZL-amenability constants of finite groups with two character degrees‎, Canad‎. ‎Math‎. ‎Bull.‎, ‎57 (2014) 449-462‎.]‎, ‎and establishing a new estimate for groups with trivial centre‎.

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References

[1] M. Alaghmandan, Y. Choi and E. Samei, ZL-amenability constants of nite groups with two character degrees, Canad. Math. Bul l., 57 (2014) 449-462.

[2] A. Azimifard, E. Samei and N. Spronk, Amenability prop erties of the centres of group algebras, J. Funct. Anal., 256 (2009) 1544-1564.

[3] P. M. Cohn, Algebra., 1, second ed., John Wiley & Sons Ltd., Chichester, 1982.

[4] P. M. Cohn, Algebra., 2, second ed., John Wiley & Sons Ltd., Chichester, 1989.
[5] P. M. Cohn, Algebra., 3, second ed., John Wiley & Sons Ltd., Chichester, 1991.

[6] I. M. Isaacs, Character theory of nite groups, Corrected reprint of the 1976 original, Academic Press, New York, Dover Publications, Inc., New York, 1994.

[7] G. James and M. Lieb eck, Representations and characters of groups, second ed., Cambridge University Press, New York, 2001.
[8] M. F. Newman, On a class of metab elian groups, Proc. London Math. Soc. (3), 10 (1960) 354-364.

[9] M. F. Newman, On a class of nilp otent groups, Proc. London Math. Soc. (3), 10 (1960) 365-375.

[10] D. Rider, Central idemp otent measures on compact groups, Trans. Amer. Math. Soc., 186 (1973) 459-479.
International Journal of Group Theory
Volume 5, Issue 4 - Serial Number 4
December 2016
Pages 27-46

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History

  • Receive Date: 19 October 2014
  • Revise Date: 16 May 2015
  • Accept Date: 19 May 2015
  • Published Online: 01 December 2016

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