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 \(n\geq 5\) ).
