On Generation of the Group $PGL_n(\mathbb{Z}+i\mathbb{Z})$ by Three Involutions, Two of which Commute

Abstract

The results of the paper relate to the following general problem. Find natural finite generating sets of elements of a given linear group over a finitely generated commutative ring. Of particular interest are coefficient rings that are generated by a single element, for example, the ring of integers or the ring of Gaussian integers. We prove that a projective general linear group of dimension $n$ over the ring of Gaussian integers is generated by three involutions two of which commute if and only if $n$ is greater than $4$ and $4$ does not divide $n$. Earlier, M.\,A.\,Vsemirnov, R.\,I.\,Gvozdev, D.\,V.\,Lev\-chuk and the authors of this paper solved a similar problem for the special and projective special linear groups.

Keywords

• projective general linear group
• the ring of gaussian integers
• generating triples of involutions