Projective Cyclic Groups in Higher Dimensions

TL;DR

This paper classifies projective transformations in complex projective space based on algebraic and dynamical properties, describing their limit sets and regions of proper discontinuity, extending concepts from CAT(0) space isometries.

Contribution

It offers a comprehensive classification of cyclic groups in PSL(n+1,C) using algebraic and dynamical methods, including descriptions of limit sets and discontinuity regions.

Findings

Classification into elliptic, parabolic, and loxodromic types.

Descriptions of Kulkarni's limit sets and regions of discontinuity.

Families of maximal regions where groups act properly discontinuously.

Abstract

In this article we provide a classification of the projective transformations in $PSL(n+1,\Bbb{C})$ considered as automorphisms of the complex projective space $\Bbb{P}^n$. Our classification is an interplay between algebra and dynamics, which just as in the case of isometries of CAT(0)-spaces, can be given by means of tree three types, namely: elliptic, parabolic and loxodromic. We carefully describe the dynamic in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, the discontinuity region and in some cases we provide families of maximal regions where the respective cyclic group acts properly discontinuously. Also we provide, in each case, some equivalents ways to classify the projective transformations.