Word maps and word maps with constants of simple algebraic groups

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 = w₁σ₁w₂ ··· w_rσ_rw_{r + 1}, w₁, …, w_{r + 1} ∈ F_m, w₂, …, w_r ≠ 1, Σ = {σ₁, …, σ_r | σ_i ∈ 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(Γ_w, G) of the group Γ_w = F_m/<w>.