Let $ p $ be a polynomial in several non-commuting variables with coefficients in a field $ K
$ of arbitrary characteristic. It has been conjectured that for any $ n $, for $ p $ multilinear …

This paper contains a survey of recent developments in the investigation of word equations
in simple matrix groups and polynomial equations in simple (associative and Lie) matrix …

The word maps w˜: GL m D 2 k→ GL n D ̃ w:\kern0. 5em GL _m (D)^ 2k → GL _n (D) and w˜:
D∗ 2 k→ D∗ ̃ w:\kern0. 5em D^ ∗ 2k → D^ ∗ for a word w=∏ i= 1 kxiyi, w= ∏ i= 1^ k\left x _i …

We extend Borel's theorem on the dominance of word maps from semisimple algebraic
groups to some perfect groups. In another direction, we generalize Borel's theorem to some …

We study the word image of words with constants in ${\rm GL}(V) $ and show that it is large
provided the word satisfies some natural conditions on its length and its critical constants …

Let $ K $ be an algebraically closed field and $\mathrm {M}(2, K) $ be the $2\times 2$ matrix
algebra over $ K $ and $\mathrm {GL}(2, K) $ be the invertible elements in $\mathrm {M}(2 …

Let K be an algebraically closed field of characteristic p≥ 0. Assume F, F (1), F (2), and (F
(1)) p are the free group on n> 1 generators, the first and second derived subgroups of F …

If $ A $ is a finite group (or a finite ring) and $\omega $ is a word map (or a polynomial map),
we define the quantity $|\omega (A)|/| A| $ as the image ratio of $\omega $ on $ A $ and will …

arXiv:2008.11773v3 [math.GR] 3 Feb 2022 Page 1 arXiv:2008.11773v3 [math.GR] 3 Feb 2022
Commutator lengths in general linear group over a skew-field. P. GVOZDEVSKY Abstract. We …

Abstract T. Bandman and Yu. G. Zarhin have proved that for every word w ∈ F_n, w ∉ F_n^
2 w∈ F n, w∉ F n 2, the corresponding word map ̃ w: PGL _2^ n (K) → PGL _2 (K) w~: PGL …