Word maps and word maps with constants of simple algebraic groups

N. L. Gordeev, B. E. Kunyavskii, E. B. Plotkin

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In the present paper, we consider word maps w: Gm → G and word maps with constants wΣ: Gm → G of a simple algebraic group G, where w is a nontrivial word in the free group Fm of rank m, wΣ = w1σ1w2 ··· wrσrwr + 1, w1, …, wr + 1 ∈ Fm, w2, …, wr ≠ 1, Σ = {σ1, …, σr | σi ∈ GZ(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 = Fm/<w>.

Original languageEnglish
Pages (from-to)632-634
Number of pages3
JournalDoklady Mathematics
Issue number3
StatePublished - 1 Nov 2016

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