Word maps, word maps with constants and representation varieties of one-relator groups

We consider word maps and word maps with constants on a simple algebraic group 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 the map induced by w∈Fm and the structure of the representation variety R(Γw,G) of the group Γw=Fm/〈w〉.