Complex Kleinian Groups

Cano, Angel; Navarrete, Juan Pablo; Seade, José

doi:10.1007/978-3-0348-0481-3_3

Cited by 17 publications

(42 citation statements)

References 14 publications

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“…If the answer to the previous question is positive, how does the dimension Hausdorff of P 2 C \ Ω(Γ) varies when we study analytic families of groups? In the case when Γ is a finite extension of a cyclic group the answer to these questions follows from (see [31,8]).…”

Section: Discontinuity Regions; the 2-dimensional Casementioning

confidence: 99%

An Overview of Complex Kleinian Groups

Cano

Seade

2015

Nonlinear Systems and Complexity

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Dedicated to Vale, on the occasion of his 65th birthday.Abstract. Classical Kleinian groups are discrete subgroups of P SL(2, C) acting on the complex projective line P 1 C , which actually coincides with the Riemann sphere, with non-empty 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 P SL(n, C), n > 2. Surprisingly, this is a branch of mathematics which is in its childhood, and in this article we give an overview of it.1991 Mathematics Subject Classification. Primary: 3nQ45, 37F45; Secondary nnE40, 57R30.

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Section: Discontinuity Regions; the 2-dimensional Casementioning

confidence: 99%

An Overview of Complex Kleinian Groups

Cano

Seade

2015

Nonlinear Systems and Complexity

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“…On the other hand it is a challenging problem to decide if a 3-manifold has a geometric structure. It is not clear when the representations obtained here are discrete (for general discrete subgroups of PGL(3, C) see [6]) . On the other hand, a criterium for rigidity of representations is given in [3] and a discussion of generic discreteness of the boundary holonomy is given in [17].…”

Section: Introductionmentioning

confidence: 98%

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

2014

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In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PGL(3, C) (in particular in PGL(2, C), PGL(3, R) and PU(2, 1)). The representations are obtained by gluing decorated tetrahedra of flags as in [10,2]. We list complete computations (giving 0-dimensional or 1-dimensional solution sets (for unipotent boundary holonomy) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the computer methods used to find them. The methods we use work for non-unipotent boundary holonomy as shown in some examples.This work was supported in part by the ANR through the project "Structures Géométriques et Triangulations".

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“…(5) If Ω Kul (Γ) = D ×C * , then Γ is controllable with infinite kernel and quasi-cocompact control group Σ such that Ω(Σ) = D. (6) If Ω Kul (Γ) = H 2 C , then Γ is contained in PU(2, 1) up to projective conjugation. To state our last theorem we recall from [24] (see also [8]) that a complex homothety is an element in PSL(3, C) that has a lift to SL(3, C) whose normal Jordan form is of the type …”

Section: Introductionmentioning

confidence: 99%

“…We refer to [8] for a thorough discussion of the subject of complex Kleinian groups and limit sets for discrete subgroups of PSL(n + 1, C).…”

Section: Introductionmentioning

confidence: 99%

On Discrete groups of automorphisms of $$\mathbb P ^2_\mathbb{C }$$

Cano

Seade

2013

Geom Dedicata

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We study the geometry and dynamics of discrete subgroups Γ of PSL(3, C) with an open invariant set Ω ⊂ P 2 C where the action is properly discontinuous and the quotient Ω/Γ contains a connected component whicis compact. We call such groups quasi-cocompact. In this case Ω/Γ is a compact complex projective orbifold and Ω is a divisible set. Our first theorem refines classical work by Kobayashi-Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds Ω/Γ. We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous.

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Complex Kleinian Groups

Cited by 17 publications

References 14 publications

An Overview of Complex Kleinian Groups

An Overview of Complex Kleinian Groups

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexity

On Discrete groups of automorphisms of $$\mathbb P ^2_\mathbb{C }$$

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