On the Generation of Certain Matrix Groups by Three Involutions, Two of which Commute

A group generated by three involutions two of which commute, is called (2×2, 2)-generated. It is known that the special linear group SLn(Z+iZ) over the ring of the Gaussian integers Z+iZ (respectively, its quotient group by the center PSLn(Z+iZ)) is (2×2, 2)-generated if and only if n > 5 and n 6= 6 (respectively, when n > 5). It is clear that the general linear group GLn(Z+iZ) is not (2×2, 2)-generated, since it contains matrices with determinant different from ±1, and the determinant of any of its involutions is equal to ±1. It is also known that the group PGLn(Z + iZ) is generated by three involutions if and only if two of them commute when n > 5 and 4 does not divide n. In this paper we consider the problem on (2 × 2, 2)-generation for the matrix group GL±1 n (Z + iZ) with determinant ±1 over the ring of the Gaussian integers and for its quotient group by the center PGL±1 n (Z + iZ).