Abstract
In the present paper, we consider word maps w: G m → G and word maps with constants w Σ: G m → G of a simple algebraic group G, where w is a nontrivial word in the free group F m of rank m, w Σ = w 1 σ 1 w 2 ··· w r σ r w r + 1, w 1, …, w r + 1 ∈ F m , w 2, …, w r ≠ 1, Σ = {σ 1, …, σ r | σ i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γw, G) of the group Γw = F m /<w>.
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Original Russian Text © N.L. Gordeev, B.E. Kunyavskii, E.B. Plotkin, 2016, published in Doklady Akademii Nauk, 2016, Vol. 471, No. 2, pp. 136–138.
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Gordeev, N.L., Kunyavskii, B.E. & Plotkin, E.B. Word maps and word maps with constants of simple algebraic groups. Dokl. Math. 94, 632–634 (2016). https://doi.org/10.1134/S1064562416060077
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DOI: https://doi.org/10.1134/S1064562416060077