Fibers of automorphic word maps and an application to composition   factors
  In this paper, we study the fibers of "automorphic word maps", a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite groups $G$ where, for some fixed nonempty reduced word $w$ in $d$ variables and some fixed $\rho\in\left(0,1\right]$, the word map $w_G$ on $G$ has a fiber of size at least $\rho|G|^d$: No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group $G$. 
Alexander Bors
math.GR