Finite geometry for a generation

There are a number of results concerning the generation of a collineation group by two of its elements. A. A. Albert and J. Thompson [1] were the first to exhibit two elements generating the little projective group PSL(d, q) of PG(d − 1, q) (for each d and q). According to a theorem of W. M. Kantor and A. Lubotzky [8], “almost every” pair of its elements generates PSL(d, q) as qd→∞ (asymptotically precise bounds on this probability are obtained in W. M. Kantor [7]). Given 1 6= g ∈ PSL(d, q), the probability that h ∈ PSL(d, q) satisfies 〈g, h〉 = PSL(d, q) was studied by R. M. Guralnick, W. M. Kantor and J. Saxl [3], and its behavior was found to depend on how qd → ∞. Yet another variation that has been proposed is “1 2 ”generation: if 1 6= g ∈ PSL(d, q) then some h ∈ PSL(d, q) satisfies 〈g, h〉 = PSL(d, q). This note concerns a stronger version of this notion: Theorem. For any d ≥ 4 and any q, there is a conjugacy class C of cyclic subgroups of PSL(d, q) such that, if 1 6= g ∈ PSL(d, q), then 〈g, C〉 = PSL(d, q) for more than ( 1− 1 q − 1 qd−1 )2 |C| elements C ∈ C. In particular, there are more than 0.4|C| such