Word Values in Finite Groups

Let w = w(x1, ..., xd) denote a group word in d variables, that is, an element of the free group of rank d. For a finite group G we may define a word map that sends a d-tuple, (g1, ..., gd) of elements of G, to its w-value, w(g1, ..., gd), by substituting variables and evaluating the word in G by performing all relevant group operations. In this thesis we study a number of problems to do with the behaviour of word maps over various classes of groups. The first problem we look at concerns the distribution of word values in nilpotent groups. We obtain a lower bound for the probability that a random d-tuple of elements from any nilpotent group of class 2 evaluates at the identity for any fixed word in d variables answering a special case of a question of Alon Amit [1]. Another problem we look at deals with the question of which possible subsets of a group can be obtained as the image of a word map. This was first studied by Kassabov & Nikolov [15] and later by Lubotzky [22] who gave a complete description in the case of simple groups. We obtain a partial classification for the almost simple groups and quasisimple groups and completely describe what happens in the case of symmetric groups. Finally, we study twisted commutator maps over the alternating groups, special linear groups and special unitary groups. Twisted commutators are similar to commutators but are twisted by group automorphisms. These have been studied by Nikolov & Segal in [29] where they obtain bounds on the width of twisted commutator words over the finite quasisimple groups. Our goal is to improve these bounds. Throughout this thesis we will also look at variations of the above problems as well as related questions.