On the surjectivity of certain word maps on SU(2)

Abstract

In this paper, we study the word map [Formula: see text] where [Formula: see text] is a word in the free group [Formula: see text] of rank 2. We give a necessary and sufficient condition for the surjectivity of the word map by using the trace polynomials. As applications of our method, we give example of word [Formula: see text] for which the word map is not surjective. For families of words belonging to [Formula: see text] of the form [Formula: see text] and [Formula: see text] we can tell whether the word map is surjective or not.