Comparison of Limit Sets for the Action of Kleinian Groups in $\mathbb{C}P^n$

TL;DR

This paper investigates various limit sets of Kleinian groups acting on complex projective spaces, establishing conditions under which different notions of limit sets coincide, especially for convex-cocompact groups.

Contribution

It compares multiple definitions of limit sets for Kleinian groups in complex projective spaces and proves their equivalence under convex-cocompactness.

Findings

Myrberg and Kulkarni limit sets coincide for convex-cocompact Kleinian groups.

Different notions of limit sets are compared and related in the context of complex projective actions.

The paper provides conditions for the equality of various limit sets in higher-dimensional projective spaces.

Abstract

We compare different notions of limit sets for the action of Kleinian groups on the $n-$dimensional projective space via the irreducible representation $\varrho:PSL(2,\mathbb{C})\to PSL(n+1,\mathbb{C}).$ In particular, we prove that if the Kleinian group is convex-cocompact, the Myrberg and the Kulkarni limit coincide.