Word Maps, Word Maps with Constants and Representation Varieties of One-relator Groups

by evaluation. Namely, w̃(g1, . . . , gm) is obtained by substituting gi in place of xi and g −1 i in place of x−1 i followed by computing the resulting value w(g1, . . . , gm). Word maps have been intensely studied over at least two past decades in various contexts (see, e.g., [?], [?], [?], [?] for surveys). In this paper, we consider the case where G = G(K) is the group of K-points of a simple linear algebraic group G defined over an algebraically closed field K. We are mainly interested in studying the image of w̃. Borel’s theorem [?] says that w̃ is dominant, i.e., its image contains a Zariski dense open subset of G. However, w̃ may not be surjective: this may happen in the case of power maps on groups with non-trivial centre (say, squaring map on SL(2,C)) and, if G is not of type A, even on adjoint groups, see [?], [?], [?]. For the adjoint groups of type A, the surjectivity problem is wide open, even in the case of groups of rank 1, and even for words in two variables. The goal of the present paper is two-fold. First, we extend our viewpoint on the dominance and surjectivity problems from genuine word maps to word maps with constants and establish a partial, “generic” analogue of Borel’s dominance theorem. Another extension concerns a continuation of the word map w̃ : GLn(K) m → GLn(K) to the map w̃∗ : Mn(K) m → Mn(K). Being interesting in its own right, this method yields, as a by-product, a new proof of some results of Bandman and Zarhin [?], who proved the surjectivity of w̃ for G = PGL2(K) in the case whereK is an algebraically closed field of characteristic zero, m = 2, and w ∈ Fm\F 2 m, where F 1 m = [Fm, Fm], . . . , F i m = [F i−1 m , F i−1 m ], . . . . Our second goal consists in studying the geometric structure of the representation variety of the one-relator group Γw := Fm/ ⟨w⟩ with an eye towards applying the data on its irreducible components to searching unipotent elements in the image of the word map. This often allows one to prove the surjectivity of the word map on PGL2(K). We give a non-trivial example of such a w ∈ F2, in the spirit of [?] but avoiding their computer calculations.