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Girth, words and diameter

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arxiv 1903.00748 v2 pith:E3PXWJRB submitted 2019-03-02 math.GR math.COmath.PR

Girth, words and diameter

Martin W. Liebeck , Aner Shalev This is my paper
classification math.GR math.COmath.PR
keywords girthcayleygraphsgroupsrandomdiameterfiniteadditional
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We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, 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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