Products of three word maps on simple algebraic groups

摘要

Let w = w(x1,..., xn) be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map w : G n. G where w((g1,..., gn)) := w(g1,..., gn) for every sequence (g1,..., gn). Let G be a simple and simply connected group which is defined and split over an infinite field K and let G = G(K). For the case when w = w1w2w3w4 and w1, w2, w3, w4 are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210: 81-100, 2015) that G Z(G). Im w where Z(G) is the center of the group G and Im w is the image of the word map w. For the case when G = SLn(K) and n = 3, in the same paper of Hui et al. (2015) it was shown that the inclusion G Z(G). Im w holds for a product w = w1w2w3 of any three non-trivial words w1, w2, w3 with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types B-2, G(2).