Asymptotic flag geometry of complex Kleinian groups in ${\mathbb{P}}_{\mathbb{C}}^2$

Abstract

We organize several natural notions of limit set for discrete subgroups of $\mathrm{PSL}(3,\mathbb{C})$ around a common asymptotic structure encoded by pseudo--projective degeneration and by the full or partial flag data carried by divergent sequences. In the $\theta$-divergent case, the two projections of the full-flag limit set recover the attracting and repelling projective boundary sets, while the Myrberg limit set is the union of the limiting projective lines. Thus the equicontinuity region is the complement of a canonical line configuration; under the usual three-line general-position hypothesis, this same configuration is the Kulkarni limit set and the equicontinuity and Kulkarni ordinary regions coincide.