Word maps of Chevalley groups over infinite fields

Let $G$ be a simply connected Chevalley group over an infinite field $K$ and let $\widetilde{w}: G^n\rightarrow G$ be a word map that corresponds to a non-trivial word $w$. It has been proved in: (Israel J. Math. 210 (2015), 81-100) that if $w = w_1w_2w_3w_4$ is a product of any four words on independent variables, then every non-central element of the group $G$ is contained in the image of $\widetilde{w}$. A similar result for a word $w = w_1w_2w_3$ that is a product of three independent words was obtained in: (Archiv der Math. 112 (2019), no. 2, 113-122) under the condition that the group $G$ is not of types $B_2, G_2$. In this paper we prove that for groups of types $B_2, G_2$ all elements of big Bruhat cell $B \mathfrak{n}_{w_0} B$ are contained in the image of a word map $\widetilde{w}$ where $w = w_1w_2w_3$ is a product of three independent words. For groups of types $A_r, C_r, G_2$ (respectively, for groups of type $A_r$) or groups over a perfect field $K$ (respectively, over a perfect field $K$ such that $\mathrm{char} K$ is not a bad prime for $G$) that has $\dim K \leq 1$ (here $\dim K$ is cohomological dimension of $K$) it has been proved here that all split regular semisimple elements (respectively, all regular unipotent elements) of the group $G$ are contained in the image $\widetilde{w}$ where $w = w_1w_2$ is a product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group $\mathcal G$ over a field $K$ of characteristic zero we show that for a word map $\widetilde{w}: \mathcal{G}(K)^n\rightarrow \mathcal{G}(K)$, where $w = w_1w_2$ is a product of two independent words, all unipotent elements are contained in $\mathrm{Im}\, \widetilde{w}$.