On the number of lines in the limit set for discrete subgroups of $PSL(3,\Bbb{C})$

TL;DR

This paper investigates the structure of limit sets for discrete subgroups of PSL(3,C), focusing on the number and configuration of complex lines in various regions related to the group's action.

Contribution

It provides a detailed analysis of the number of complex lines in the limit set and conditions under which the equicontinuity and Kulkarni's regions coincide for subgroups of PSL(3,C).

Findings

Determines the number of complex lines in the limit set.

Establishes conditions for the equicontinuity set to match Kulkarni's region.

Describes the limit set in terms of group elements.

Abstract

Given a discret subgroup $\Gamma\subset PSL(3,\C)$, we determine the number of complex lines and complex lines in general position lying in the complement of: maximal regions on which $\Gamma$ acts properly discontinuously, the Kularni's limit set of $\Gamma$ and the equicontinuity set of $\Gamma$. We also provide sufficient conditions to ensure that the equicontinuity region agrees with the Kulkarni's discontinuity region and is the largest set where the group acts properly discontinuously and we provide a description of he respective limit set in terms of the elements of the group.