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

The Waring problem for finite simple groups

“…[EG98] and the references therein), establish this for alternating groups and finite simple groups of Lie type defined over finite fields of large enough size. More recent related results can be found in [Sha09] and [Sha08]. In this section we prove Theorem 1.1.4, which may be regarded as an asymptotic approximation of Thompson's conjecture.…”

“…It is shown in [Sha09] (see also [Sha08]) that if Γ is a finite simple group of Lie type, then |w(Γ) 2 |/|Γ| → 1 as |Γ| → ∞. Larsen and Shalev then showed in [LS09] that if w = 1 and the simple group Γ is alternating, or of Lie type of bounded rank (excluding Suzuki groups and Ree groups), then there exists a number N w depending on w (and in the latter case also on the bound on the rank of Γ) such that |Γ| ≥ N w implies w(Γ) 2 = Γ.…”

“…Indeed Shalev [Sha09] showed that for any nontrivial word w, there exists a number N w such that if Γ is a finite simple group of order at least N w , then…”

The Ore conjecture

“…Shalev [41] proved that if g is a random element of a finite simple group G, then the probability that g is a commutator tends to 1 as |G| → ∞. This implies that every element of a large finite simple group is a product of two commutators.…”

On the shortest identity in finite simple groups of Lie type

“…Recently there has been great interest in the study of word maps in finite simple groups and in algebraic groups (see [10], [11], [17], [19]). Let w ¼ wðx 1 ; .…”