Surjectivity of certain word maps on $${{\mathrm{PSL}}}(2,\mathbb {C})$$ PSL ( 2 , C ) and $${{\mathrm{SL}}}(2,\mathbb {C})$$ SL ( 2 , C )

Abstract

Let $$n \geqslant 2$$ be an integer and $$F_n$$ the free group on n generators, its first and second derived subgroups. Let K be an algebraically closed field of characteristic zero. We show that if , then the corresponding word map is surjective. We also describe certain word maps that are surjective on $${{\mathrm{SL}}}(2,\mathbb {C})$$ .