Diameters of random Cayley graphs of finite nilpotent groups

El-Baz, D. and Pagano, C. (2021) Diameters of random Cayley graphs of finite nilpotent groups. Journal of Group Theory, 24(5), pp. 1043-1053. (doi: 10.1515/jgth-2020-0066)

Abstract

We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Pagano, Dr Carlo
• Authors: El-Baz, D., and Pagano, C.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Group Theory
• Publisher: De Gruyter
• ISSN: 1433-5883
• ISSN (Online): 1435-4446
• Published Online: 02 February 2021
• Copyright Holders: Copyright © 2021 de Gruyter
• First Published: First published in Journal of Group Theory 24(5): 1043-1053
• Publisher Policy: Reproduced in accordance with the publisher copyright policy