Juan Pablo Navarrete

Complex Kleinian Groups

In this chapter we introduce some fundamental concepts in the theory of complex Kleinian groups that we study in the sequel. We begin with an example in \(\mathbb{P}^{2}_\mathbb{C}\) that illustrates the diversity of possibilities one has when defining the notion of “limit set”. In this example we see that there are several nonequivalent such notions, each having its own interest.

Complex Schottky Groups

Classical Schottky groups in PSL(2, C) play a key role in both complex geometry and holomorphic dynamics. On one hand, Kobe’s retrosection theorem says that every compact Riemann surface can be obtained as the quotient of an open set in the Riemann sphere S2 which is invariant under the action of a Schottky group. On the other hand, the limit sets of Schottky groups have rich and fascinating geometry and dynamics, which has inspired much of the current knowledge we have about fractal sets and 1-dimensional holomorphic dynamics.

On the number of lines in the limit set for discrete subgroups of PSL(3, ℂ)

Given a discrete subgroup G PSL.3;C/, acting on the complex projective plane, P 2 C , in the canonical way, we list all possible values for the number of complex projective lines and for the maximum number of complex projective lines lying in the complement of each of: the equicontinuity set of G, the Kulkarni discontinuity region of G, and maximal open subsets of P 2 C on which G acts properly discontinuously.

Subgroups of (3,ℂ) with four lines in general position in its limit set

In this article we provide an algebraic characterization of those groups of

Kleinian Groups and Twistor Theory

PSL(3,\Bbb{C})

Kleinian Groups with a Control Group

whose limit set in the Kulkarni sense has, exactly, four lines in general position. Also we show that, for this class of groups, the equicontinuity set of the group is the largest open set where the group acts discontinuously and agrees with the discontinuity set of the group.

Projective Orbifolds and Dynamics in Dimension 2

Twistor theory is one of the jewels of mathematics in the 20th Century. A starting point of the celebrated “Penrose twistor programme” is that there is a rich interplay between the conformal geometry on even-dimensional spheres and the holomorphic on their twistor spaces. Here we follow [202] and explain how the relations between the geometry of a manifold and the geometry of its twistor space, can be carried forward to dynamics. In this way we get that the dynamics of conformal Kleinian groups embeds in the dynamics of complex Kleinian groups.

Geometry and Dynamics of Automorphisms of \mathbb{P}^{2}_\mathbb{C}

Nowadays the literature and the knowledge about subgroups of PSL(2, \(\mathbb{C}\)) is vast. Thence, when going up into higher dimensions, a natural step is to consider Kleinian subgroups of PSL(3, \(\mathbb{C}\)) whose geometry and dynamics are “governed” by a subgroup of PSL(2, \(\mathbb{C}\)). That is the subject we address in this chapter. The corresponding subgroup in PSL(2 ,\(\mathbb{C}\)) is the control group. These groups play a significant role in the classification theorems we give in Chapter 8.

Complex Hyperbolic Geometry

Kobe’s retrosection theorem says that every compact Riemann surface is isomorphic to an orbit space Ω/Ґ, where Ω is an open set in the Riemann sphere S2 = PC and Ґ is a discrete subgroup of PSL(2,C) that leaves Ω invariant; in fact Γ is a Schottky group.

On the Dynamics of Discrete Subgroups of PU(n, 1)

In this chapter we study and describe the geometry, dynamics and algebraic classification of the elements in PSL(3, \(\mathbb{C}\)), extending Goldman’s classification for the elements in PU(2, 1) ⊂ PSL(3,\(\mathbb{C}\)). Just as in that case, and more generally for the isometries of manifolds of negative curvature, the automorphisms of \(\mathbb{P}^{2}_\mathbb{C}\) can also be classified into the three types of elliptic, parabolic and loxodromic (or hyperbolic) elements, according to their geometry and dynamics. This classification can be also done algebraically, in terms of their trace.