Projective cyclic groups in higher dimensions

Abstract

In this article we provide a classification of the projective transformations in P SL(n + 1, C) considered as automorphisms of the complex projective space P n C . Our classification is an interplay between algebra and dynamics. Just as in the case of isometries of CAT (0)-spaces, this is given by means of three types of transformations, namely: elliptic, parabolic and loxodromic. We describe the dynamics in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, minimal sets, the discontinuity region and maximal regions where the corresponding cyclic group acts properly discontinuously. We also provide, in each case, some equivalent ways to classify the projective transformations.

A Family of Complex Kleinian Split Solvable Groups

“…[1] [3]) also every discontinuity region of an infinite complex Kleinian group contains a complex projective line while in the classical theory there are groups with only finitely many points in the complement of the discontinuity region. In dimension N > 2 there are a lot of technical difficulties to determine whether a given discrete subgroup of PSL(N + 1, C) is complex Kleinian, for example the Kulkarni limit set, one of the most important tools in dimension 2, is hard to compute [4,6,5]. In this article we aim to propose a new approach in order to construct a discontinuity region for a family of discrete subgroups of PSL(N + 1, C).…”

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

“…where k; l ∈ K and -: G → a + : The mapis known as the Cartan projection of g ; and by the KAK-decomposition to the product in (7).…”

“…Other examples are discrete subgroups of PSL(3; C) whose action on CP 2 is irreducible, see [4], or when the subgroup is virtually solvable, see [2]. Notice that in dimensions bigger that two there no known examples beside the complex hyperbolic case and cyclic subgroups of PSL(3; C) (see [7]), for this reason it is important to provide new examples.…”

“…Recall that every proximal element of PSL(n + 1; C) is loxodromic according to the classification given on [8]. Definition 1.9.…”

On Classical Uniformization Theorems for Higher Dimensional Complex Kleinian Groups

“…A γ −1 for some pairwise disjoint, non-empty compact sets of P n C we deduce that γ has at least two fixed points. Therefore γ has a lift γ ∈ SL(n + 1, C) whose normal Jordan form is, see [4,9]:…”