Diameters of Finite Simple Groups: Sharp Bounds and Applications

Abstract

Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph Γ(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of arbitrary order). We also show that for any word w = w(x 1 , . . . , x d ), there is a constant c = c(w) such that for any simple group G on which w does not vanish, every element of G is a product of c values of w. From this we deduce that every verbal subgroup of a semisimple profinite group is closed. Other applications concern covering numbers, expanders, and random walks on finite simple groups.

Word maps, conjugacy classes, and a noncommutative Waring-type theorem

“…Let us now consider conjugacy classes of randomly chosen elements. It is shown in Theorem 1.12 of [19] that there exists an absolute constant c, such that if G is a finite simple group, and x 2 G is chosen at random, then we have .x G / c D G with probability tending to 1 as jGj ! 1.…”

“…For more background on products of conjugacy classes in finite simple groups, see [1], [19], and the references therein.…”

The Waring problem for finite simple groups

“…Extending the aforementioned results on powers and commutators, Liebeck and Shalev [LS01] showed in 2001 that for any word w there exists a positive integer c w depending on w such that if Γ is a finite simple group and w(Γ) = {1}, then w(Γ) cw = Γ. No explicit bounds on c w were given.…”

Commutator maps, measure preservation, and 𝑇-systems

“…. , x m ) ∈ F m gives rise to a word map α w : G m → G. Word maps on algebraic groups and on finite simple groups have been the subject of active investigations in recent years; see [Bo], [LiSh2], [La], [Sh] and [LaSh].…”