Angel Cano

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

Action of R-Fuchsian groups on CP2.

PSL(3,\Bbb{C})

An Overview of Complex Kleinian Groups

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.

Kleinian Groups and Twistor Theory

We look at lattices in Iso+(H2R)Iso+(HR2), the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on CP2CP2 via the natural embedding of SO+(2,1)↪SU(2,1)⊂SL(3,C)SO+(2,1)↪SU(2,1)⊂SL(3,C). We use the Hermitian cross product in C2,1C2,1 introduced by Bill Goldman, to determine the topology of the Kulkarni limit set ΛKulΛKul of these lattices, and show that in all cases its complement ΩKulΩKul has three connected components, each being a disc bundle over H2RHR2. We get that ΩKulΩKul coincides with the equicontinuity region for the action on CP2CP2. Also, it is the largest set in CP2CP2 where the action is properly discontinuous and it is a complete Kobayashi hyperbolic space. As a byproduct we get that these lattices provide the first known examples of discrete subgroups of SL(3,C)SL(3,C) whose Kulkarni region of discontinuity in CP2CP2 has exactly three connected components, a fact that does not appear in complex dimension 11 (where it is known that the region of discontinuity of a Kleinian group acting on CP1CP1 has 00, 11, 22 or infinitely many connected components).

Kleinian Groups with a Control Group

Classical Kleinian groups are discrete subgroups of \(PSL(2,{\mathbb{C}})\) acting on the complex projective line \(\mathbb{P}_\mathbb{C}^1\) (which coincides with the Riemann sphere) with nonempty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group \(PSL(n,{\mathbb{C}})\), \(n> 2\). Surprisingly, this is a branch of mathematics which is in its childhood, and in this chapter we give an overview of it.

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.