On the Generation of the Groups $$\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$$ and $$\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$$ by Three Involutions Two of Which Commute. II

Abstract

We complete the solution of the problem on the existence of generating triplets of involutions two of which commute for the special linear group \(\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})\) and the projective special linear group \(\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})\) over the ring of Gaussian integers. The answer has only been unknown for \(\mathrm{SL}_5\), \(\mathrm{PSL}_6\), and \(\mathrm{SL}_{10}\). We explicitly indicate the generating triples of involutions in these three cases, and we make a significant use of computer calculations in the proof. Taking into account the known results for the problem under consideration, as a consequence, we obtain the following two statements. The group \(\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})\) (respectively, \(\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})\)) is generated by three involutions two of which commute if and only if \(n\geq 5\) and \(n\neq 6\) (respectively, if \(n\geq 5\)).