On the diameter of Cayley graphs of classical groups with generating sets containing a transvection

Garonzi, Martino and Halasi, Zoltán and Somlai, Gábor (2023) On the diameter of Cayley graphs of classical groups with generating sets containing a transvection. ISRAEL JOURNAL OF MATHEMATICS. ISSN 0021-2172

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Abstract

A well-known conjecture of Babai states that if G is any finite simple group and X is a generating set for G, then the diameter of the Cayley graph Cay(G, X) is bounded by log |G|c for some universal constant c. In this paper, we prove such a bound for Cay(G, X) for G = PSL(n, q), P Sp(n, q) or P SU (n, q) where q is odd, under the assumptions that X contains a transvection and q 6 = 9 or 81.

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