Girth, words and diameter

Martin W. Liebeck; Aner Shalev

doi:10.1112/blms.12251

Girth, words and diameter

Martin W. Liebeck, Aner Shalev

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We study the girth of Cayley graphs of finite classical groups (Formula presented.) on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word (Formula presented.) takes the value 1 when evaluated in (Formula presented.) in terms of the length of (Formula presented.), which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.

Original language	American English
Pages (from-to)	539-546
Number of pages	8
Journal	Bulletin of the London Mathematical Society
Volume	51
Issue number	3
DOIs	https://doi.org/10.1112/blms.12251
State	Published - Jun 2019

Bibliographical note

Funding Information:
Received 15 November 2018; published online 10 April 2019. 2010 Mathematics Subject Classification 20D06 (primary), 20P05, 05C80, 05C12 (secondary). We are grateful to Sean Eberhard for some enlightening comments. The second author acknowledges the support of ISF grant 686/17, BSF grant 2016072 and the Vinik chair of mathematics which he holds.

Publisher Copyright:
© 2019 London Mathematical Society

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10.1112/blms.12251

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abstract = "We study the girth of Cayley graphs of finite classical groups (Formula presented.) on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word (Formula presented.) takes the value 1 when evaluated in (Formula presented.) in terms of the length of (Formula presented.), which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.",

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N2 - We study the girth of Cayley graphs of finite classical groups (Formula presented.) on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word (Formula presented.) takes the value 1 when evaluated in (Formula presented.) in terms of the length of (Formula presented.), which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.

AB - We study the girth of Cayley graphs of finite classical groups (Formula presented.) on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word (Formula presented.) takes the value 1 when evaluated in (Formula presented.) in terms of the length of (Formula presented.), which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.

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Girth, words and diameter

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