Related papers: On the Surjectivity of Engel Words on PSL(2,q)

We show that an element w of a free group F on n generators defines a surjective word map of PSL(2,C)^n onto PSL(2,C) unless w belongs to the second derived subgroup of F. We also describe certain words maps that are surjective on SL(2,C) x…

We determine the positive integers a,b and the prime powers q for which the word map w(x,y)=x^ay^b is surjective on the group PSL(2,q) (and SL(2,q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2,q)…

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map sending (x,y) to the product of the Nth powers of x and y is…

Jambor--Liebeck--O'Brien showed that there exist non-proper-power word maps which are not surjective on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$ for infinitely many $q$. This provided the first counterexamples to a conjecture of Shalev which…

In this article, we show the surjectivity of word maps w from SU(2)* SU(2) to SU(2) induced by several families of words in the free group of rank 2. Also, we prove the surjectivity of certain word maps on SL(2,C).

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 words with constants. We also consider the surjectivity problem for…

Let $w = [[x^k, y^l], [x^m, y^n]]$ be a non-trivial double commutator word. We show that $w$ is surjective on $\mathrm{PSL}_2(K)$, where $K$ is an algebraically closed field of characteristic $0$.

We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.

Given a finite group $G$, the Engel graph of $G$ is a directed graph encoding pairs of elements satisfying some Engel word. From the work of Detomi, Lucchini and Nemmi, the strongly connectivity of the Engel graph of an arbitrary group $G$…

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is said to be concise if w(G) is finite whenever the set of w-values in G is finite. In the sixties P. Hall asked whether…

The Engel graph of a finite group $G$ is a directed graph encoding the pairs of elements in $G$ satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of…

Let F be the free group on two letters. For {\omega} \in F we study the associated word map {\omega}: SU(n) \times SU(n) \rightarrow SU(n). Extending a method of Goto, we show that for {\omega} not in the second derived subgroup F^(2) of F,…

We construct non-power words which have small image in SL(2; 22n) for each n. In particular, the corresponding word maps are non-surjective. We also use this to construct word maps whose values are precisely the identity and a single…

Let $\mathbf{F}$ be the free group on two generators $a, b$ and let a family of words $w = [[a, b], [a^3, b^n]]$ in $\mathbf{F}$. In this paper we examine surjectivity of word map $w$ on special unitary group SU(2) over complex field…

A group word $w$ is said to be strongly concise in a class $\mathscr C$ of profinite groups if, for any group $G$ in $\mathscr C$, either $w$ takes at least continuum values in $G$ or the verbal subgroup $w(G)$ is finite. It is conjectured…

Word maps provide a wealth of information about finite groups. We examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group. We show that a finite group is nilpotent if…

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. There are various consequences: We prove that for every…

For any positive integers m and n, the word map (x,y) -> x^m y^n is almost measure preserving on large finite simple groups G.

In the paper we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that if $G\simeq \PSL_n^\varepsilon (q)$, then for every power words of type $x^M$ there exist constant…

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is called concise if w(G) is finite whenever the set of w-values in G is finite. It is an open question whether every word is…