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Archiv der Mathematik

, Volume 112, Issue 2, pp 113–122 | Cite as

Products of three word maps on simple algebraic groups

  • E. Egorchenkova
  • N. GordeevEmail author
Article
  • 16 Downloads

Abstract

Let \(w = w(x_1, \ldots , x_n)\) be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map \(\widetilde{w}: G^n\rightarrow G\) where \(\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)\) for every sequence \((g_1, \ldots , g_n)\). Let \(\mathcal G\) be a simple and simply connected group which is defined and split over an infinite field K and let \(G =\mathcal G(K)\). For the case when \(w = w_1w_2 w_3 w_4\) and \(w_1, w_2, w_3, w_4\) are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that \(G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}\) where Z(G) is the center of the group G and \({{\text { Im}}}\, {\widetilde{w}}\) is the image of the word map \(\widetilde{w}\). For the case when \(G = {{\text {SL}}}_n(K)\) and \(n \ge 3\), in the same paper of Hui et al. (2015) it was shown that the inclusion \(G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}\) holds for a product \(w = w_1w_2 w_3\) of any three non-trivial words \( w_1, w_2, w_3\) 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 \(B_2, G_2\).

Keywords

Word maps Simple algebraic groups 

Mathematics Subject Classification

Primary 20G15 Secondary 20G07 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsHerzen State Pedagogical UniversitySt. PetersburgRussia

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