Products of three word maps on simple algebraic groups

Abstract

Let $w = w(x_1, \ldots , x_n)$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $\widetilde{w}: G^n\rightarrow G$ where $\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$ for every sequence $(g_1, \ldots , g_n)$. Let $\mathcal G$ be a simple and simply connected group which is defined and split over an infinite field K and let $G =\mathcal G(K)$. For the case when $w = w_1w_2 w_3 w_4$ and $w_1, w_2, w_3, w_4$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$ where Z(G) is the center of the group G and ${{\text { Im}}}\, {\widetilde{w}}$ is the image of the word map $\widetilde{w}$. For the case when $G = {{\text {SL}}}_n(K)$ and $n \ge 3$, in the same paper of Hui et al. (2015) it was shown that the inclusion $G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$ holds for a product $w = w_1w_2 w_3$ of any three non-trivial words $w_1, w_2, w_3$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $B_2, G_2$.