An element w in the free group on r letters defines a map f from G^r to G for each group G. In this note, we show that whenever w is non-trivial and G is a semisimple algebraic group, f is dominant. When G is a finite simple group, the ratio \log |f(G^r)| over \log |G| tends to 1 as the order of G tends to infinity.
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 $0$.
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 stated that if a two-variable word is not a proper power of a non-trivial word, then the corresponding word map is surjective on $\mathrm{PSL}_2(\mathbb{F}_{q})$ for all sufficiently large $q$. Motivated by their work, we construct new examples of these types of non-surjective word maps. As an application, we obtain non-surjective word maps on the absolute Galoi…
It was shown by Lubotzky in 2014 that automorphism invariant subsets of finite simple groups which contain identity are always word images. In this article, we study word maps on finite nilpotent groups and show that for arbitrary finite groups, the number of automorphism invariant subsets containing identity which are not word images, referred to as word image impostors, may be arbitrarily larger than the number of actual word images. In the course of it, we construct a $2$-exhaustive set of word maps on nilpotent groups of class $2$ and demonstrate its minimality in some cases.
We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. Given a word $w$ in a free Lie algebra $\mathcal{L}_{r}$, it induces a word map $φ_{w}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ for every semisimple Lie algebra $\mathfrak{g}$. Given two words $w_{1}\in\mathcal{L}_{r_{1}}$ and $w_{2}\in\mathcal{L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $φ_{w_{1}}*φ_{w_{2}}:=φ_{w_{1}}+φ_{w_{2}}:\mathfrak{g}^{r_{1}+r_{2}}\rightarrow\mathfrak{g}$. We show that …
