A 2-dimensional Complex Kleinian Group With Infinite Lines in the Limit Set Lying in General Position

TL;DR

This paper constructs a specific 2-dimensional complex Kleinian group acting on projective space with a limit set containing infinitely many lines in general position, demonstrating new complex dynamics outside classical hyperbolic groups.

Contribution

It provides an explicit example of a discrete group with a complex limit set featuring infinitely many lines in general position, not conjugate to known complex hyperbolic groups.

Findings

The group acts on projective space with no invariant subspaces.

Its limit set contains infinitely many lines in general position.

The group is not conjugate to complex hyperbolic groups.

Abstract

In this article we present an example of a discrete group $\Sigma_\C\subset PSL(3,\Bbb{R})$ whose action on $\P^2$ does no have invariant projective subspaces, is not conjugated to complex hyperbolic group and its limit set in the sense of Kulkarni on $\Bbb{P}^2_\Bbb{C}$ has infinite lines in general position.