One line complex Kleinian groups (Q2256290)

It is well-known that Kleinian groups are discrete subgroups of \(\mathrm{PSL}(2,\mathbb{C})\) acting on \(P_\mathbb{C}^1\). They are classified as elementary groups, whose limit set is equal to zero, one, or two points, or nonelementary groups, whose limit set contains more than two points. This paper concentrates on complex Kleinian groups, discrete subgroups of \(\mathrm{PGL}(3,\mathbb{C})\) acting properly and discontinuously on some open subset of \(P_\mathbb{C}^2\). In this setting, the Kulkarni limit set is an analogue to the limit set of Kleinian groups. It is proven that the Kulkarni limit set contains a complex project line whenever the group is infinite; in more cases, the Kulkarni limit set is a union of complex projective lines. A group is elementary if the Kulkarni limit set consists of a finite union of complex projective lines. The present paper's main result gives an algebraic description of groups with the Kulkarni limit set equal to one complex project line.