Word maps and word maps with constants of simple algebraic groups
- Authors: Gordeev N.L.1,2, Kunyavskii B.E.3, Plotkin E.B.3
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Affiliations:
- Herzen State Pedagogical University of Russia
- St. Petersburg State University
- Bar-Ilan University
- Issue: Vol 94, No 3 (2016)
- Pages: 632-634
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/224487
- DOI: https://doi.org/10.1134/S1064562416060077
- ID: 224487
Cite item
Abstract
In the present paper, we consider word maps w: Gm → G and word maps with constants wΣ: Gm → G of a simple algebraic group G, where w is a nontrivial word in the free group Fm of rank m, wΣ = w1σ1w2 ··· wrσrwr + 1, w1, …, wr + 1 ∈ Fm, w2, …, wr ≠ 1, Σ = {σ1, …, σr | σi ∈ 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(Γw, G) of the group Γw = Fm/<w>.
About the authors
N. L. Gordeev
Herzen State Pedagogical University of Russia; St. Petersburg State University
Author for correspondence.
Email: nickgordeev@mail.ru
Russian Federation, St. Petersburg; St. Petersburg
B. E. Kunyavskii
Bar-Ilan University
Email: nickgordeev@mail.ru
Israel, Ramat Gan
E. B. Plotkin
Bar-Ilan University
Email: nickgordeev@mail.ru
Israel, Ramat Gan