Triangle Groups as Subgroups of Unitary Groups

This paper presents a new approach to (2, 3, k)-generated groups. We first show that the triangle group T(2, 3, k) is isomorphic to a subgroup of a two-dimensional projective unitary group over a ring. Then our main purpose is to determine the values of k for which this subgroup coincides with the whole projective unitary group. We answer this question in full generality, showing that equality holds precisely when k {2, 3, 4, 5, 7, 9, 11}. The methods come from algebraic number theory. The computer packages PARI-GP and GAP were used in this research.