How random are word maps?
Michael Larsen · 2014
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 x 1 , . . . , 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 x 1 , . . . , x d respectively the coordinates g 1 , . . . , 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.