Images of word maps in finite simple groups
  In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset $A$ of a finite simple group $G$ is the image of some word map $w : G\times G\to G $ if and only if (i) $A$ contains the identity and (ii) $A$ is invariant under $\Aut (G)$. 
Alexander Lubotzky
math.GR