Surjectivity of word maps on special linear groups of degree 2

Abstract

Let K be an algebraically closed field of characteristic p≥0. Assume F, F(1), F(2), and (F(1))p are the free group on n>1 generators, the first and second derived subgroups of F, and the subgroup of F(1) generated by all p-powers, respectively. In this paper, among other results, we show that for any word w∈F(1)∖F(2)(F(1))p, the corresponding word map w:∏nPSL2(K)→PSL2(K) is surjective. We show that the class of such words w contains, for instance, all (weakly) Engel words.