Word maps and word maps with constants of simple algebraic groups

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 (Γ w , G ) of the group Γ w = F m /< w >.