Word Maps of Chevalley Groups Over Infinite Fields

Let G be a simply connected Chevalley group over an infinite field K, and let $\tilde{w}$ : Gⁿ → G be a word map that corresponds to a nontrivial word w. In 2015, it has been proved that if w = w₁w₂w₃w₄ is the product of four words in independent variables, then every noncentral element of G is contained in the image of $\tilde{w}$. A similar result for a word w = w₁w₂w₃, which is the product of three independent words, was obtained in 2019 under the condition that the group G is not of type B₂ or G₂. In the present paper, it is proved that for a group of type B₂ or G₂, all elements of the large Bruhat cell B n_w0B are contained in the image of the word map $\tilde{w}$, where w = w₁w₂w₃ is the product of three independent words. For a group G of type A_r, C_r, or G₂ (respectively, for a group of type A_r) or a group over a perfect field K (respectively, over a perfect field K the characteristic of which is not a bad prime for G) with dim K ≤ 1 (here, dim K is the cohomological dimension of K), it is proved that all split regular semisimple elements (respectively, all regular unipotent elements) of G are contained in the image of $\tilde{w}$, where w = w₁w₂ is the product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group G over a field K of characteristic zero, it is shown that for a word map $\tilde{w}$ : G(K)ⁿ → G(K), where w = w₁w₂ is a product of two independent words, all unipotent elements are contained in Im $\tilde{w}$.