Complex Kleinian Groups

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

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

Cited by 25 publications

(23 citation statements)

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“…Unlike Proposition 3.4, we don't need to state any conditions on · . We need only that G(z) = T −a S(z) for all z ∈ a , which is true by (5).…”

Section: The Finite Building Propertymentioning

confidence: 99%

Finite Partitions for Several Complex Continued Fraction Algorithms

Abrams

2020

Experimental Mathematics

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We present a property satisfied by a large variety of complex continued fraction algorithms (the "finite building property") and use it to explore the structure of bijectivity domains for natural extensions of Gauss maps. Specifically, we show that these domains can each be given as a finite union of Cartesian products in C × C. In one complex coordinate, the sets come from explicit manipulation of the continued fraction algorithm, while in the other coordinate the sets are determined by experimental means.2 ), respectively. In [15], Katok-Ugarcovici use the natural extension to calculate the invariant measure for any (a, b)-continued fraction Gauss map. In the complex setting, this method was applied by Tanaka [20] to the nearest even integer algorithm and by Ei et al. [7] to the nearest integer algorithm.In this paper, we investigate complex continued fractions and their Gauss maps' natural extensions G acting on C × C. In particular, we describe a substitute for Date: 23 October 2019.

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“…Unlike Proposition 3.4, we don't need to state any conditions on · . We need only that G(z) = T −a S(z) for all z ∈ a , which is true by (5).…”

Section: The Finite Building Propertymentioning

confidence: 99%

Finite Partitions for Several Complex Continued Fraction Algorithms

Abrams

2020

Experimental Mathematics

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“…Discrete groups of projective transformations arise as monodromy groups of ordinary differential equations (see [9]), associated to Ricatti 's foliations (see [12]) or as the monodromy groups of the so called orbifold uniformizing differential equations (see [14]). However outside the groups coming from complex hyperbolic geometry, little is known about their dynamics, see [3]. Yet, as in the one dimensional case, one might expect interesting results.…”

Section: Introductionmentioning

confidence: 99%

Projective cyclic groups in higher dimensions

Cano

Loeza

Ucan-Puc

2017

Linear Algebra and its Applications

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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.

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“…Back in the 1990s, Seade and Verjovsky began the study of discrete groups acting on projective spaces, see [13][14][15]. Over the years, new results have been discovered, see [4]. However, it has been hard to construct groups acting on P 2 C which are neither virtually affine nor complex hyperbolic.…”

Section: Introductionmentioning

confidence: 99%

“…Kulkarni's limit set has the following properties. For a more detailed discussion of this in the two-dimensional setting, see [4]. Proposition 1.4 (See [1,4,5,11]).…”

Section: Introductionmentioning

confidence: 99%

“…As a corollary we get that every discrete compact surface group in PO + (2, 1) admits a deformation in PSL(3, C) with a non-empty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on P 2 C , see [4,6,[13][14][15].Partially supported by grants of the PAPPIT's project IA100112.…”

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confidence: 99%

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Two-dimensional Veronese groups with an invariant ball

Cano

Loeza

2017

Int. J. Math.

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Abstract. In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in P 2 C . As a corollary we get that every discrete compact surface group in PO + (2, 1) admits a deformation in PSL(3, C) with a non-empty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on P 2 C , see [4,6,[13][14][15].

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Complex Kleinian Groups

Cited by 25 publications

References 0 publications

Finite Partitions for Several Complex Continued Fraction Algorithms

Finite Partitions for Several Complex Continued Fraction Algorithms

Projective cyclic groups in higher dimensions

Two-dimensional Veronese groups with an invariant ball

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