Surjectivity of Engel Words on ${\mathrm{SL}}_2(\mathcal{O})$ and ${\mathrm{PSL}}_2({\mathcal{O}}_2)$

Abstract

The study of word maps and Waring-like problems has been widely pursued for finite simple groups, algebraic groups, and Lie groups. In this article, we study Engel word maps $e_m(x,y)=\left[\cdots \left[[x,y],y\right],\cdots ,y\right]$ on certain linear groups over local rings, namely, ${\mathrm{SL}}_2(\mathcal{R})$ and ${\mathrm{PSL}}_2(\mathcal{R})$. We consider the commutative ring $\mathcal{R}$ to be either a complete, local principal ideal ring $\mathcal{O}$, or a local principal ideal ring of finite length ${\mathcal{O}}_{\ell }$. Suppose the characteristic of the residue field $k\cong {\mathbb{F}}_q$ is $\ne 2$. Under some mild conditions on $q$, we show that there exists a constant $q_0(m)$, such that for all $q\ge q_0(m)$, all lifts in ${\mathrm{SL}}_2(\mathcal{O})$ of non-scalar elements of ${\mathrm{SL}}_2(k)$, are in the image of the $m$-th Engel word over ${\mathrm{SL}}_2(\mathcal{O})$. We further show that all Engel word maps are surjective on ${\mathrm{PSL}}_2({\mathcal{O}}_2)$ where ${\mathcal{O}}_2$ is a local principal ideal ring of length two. This work generalizes similar results about the Engel word map over fields.

Version 1, 35 pages, Comments are welcome