How random are word maps?

Summary

Any word ω in the free group on d generators determines a function G^d → G for every group G. If ω is a fixed nontrivial word and G ranges over the finite simple groups, the resulting sequence of functions can be expected to enjoy some properties of a random sequence of functions. In this paper, we review the current state of the art, emphasizing open problems.

Let F_d denote the free group on d generators x₁, . . . , x_d and ω ∊ F_d a nontrivial element. For every group G, w defines a word map f ω G:Gd !G obtained by substituting for x₁, . . . , x_d respectively the coordinates g₁, . . . , g_d of a given element of G_d. A number of authors have examined the behavior of ƒ _w,G when ƒis fixed and G ranges over some set of groups, especially the set of all (nonabelian) finite simple groups. A unifying theme behind a good deal of recent work is this question: for a given word ƒ, do the maps ƒ ƒ G behave like random functions G^d → G? The answer depends partly on the choice of w but also on what properties of random functions are desired. This paper examines some recent progress in understanding basic randomness properties of word maps, with an emphasis on open questions.