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Angel Cano

Publications and source records attributed to Angel Cano.

16 recordsLinked to original sources
arXiv2026

Asymptotic flag geometry of complex Kleinian groups in $\mathbb{P}^2_\mathbb{C}$

We organize several natural notions of limit set for discrete subgroups of $\mathrm{PSL}(3,\mathbb{C})$ around a common asymptotic structure encoded by pseudo--projective degeneration and by the full or partial flag data carried by divergent sequences. In the $\theta$-divergent case, the two projections of the full-flag limit set recover the attracting and repelling projective boundary sets, while the Myrberg limit set is the union of the limiting projective lines. Thus the equicontinuity region is the complement of a canonical line configuration; under the usual three-line general-position hypothesis, this same configuration is the Kulkarni limit set and the equicontinuity and Kulkarni ordinary regions coincide.

math.DS↗
arXiv2023

Elementary groups in $\PSL(3,\C)$

In this paper, we give a classification of the subgroups of $\textrm{PSL}(3, \mathbb{C})$ that act on $\mathbb{P}_{\mathbb{C}}^2$ in such a way that their Kulkarni limit set has finitely many lines in general position lines. These are the elementary groups.

math.GR↗
arXiv2023

Exceptional algebraic curves for infinite groups of $\textrm{PGL}(n,\mathbb{C})$

We classify algebraic curves in $\mathbb{CP}^{n}$ ($n \geq 2$) that are invariant under an infinite subgroup of $\operatorname{PGL}(n+1,\mathbb{C})$. In particular, we prove that any irreducible, non-degenerate, one-dimensional algebraic set in $\mathbb{CP}^{n}$ invariant under an infinite subgroup of $\operatorname{Aut}(\mathbb{CP}^{n})$ must be projectively equivalent to a monomial curve.

math.AG↗
arXiv2021

Complete hyperbolic structures in the complement of the Borromean rings

In this note, we show the fundamental group of the complement of the Borromean rings in $\Bbb{S}^3$ has exactly two representations in ${\rm PSL}(2,\Bbb{C})$ which are faithful, discrete and send meridians into parabolic elements. Using this result we are able to show that Borromean rings admit exactly one hyperbolic structure.

math.GT↗
arXiv2018

Discrete parabolic groups in ${\rm PSL}(3, \Bbb{C})$

We study and classify the purely parabolic discrete subgroups of $PSL(3,\Bbb{C})$. This includes all discrete subgroups of the Heisenberg group ${\rm Heis}(3,\Bbb{C})$. While for $PSL(2,\Bbb{C})$ every purely parabolic subgroup is Abelian and acts on $\Bbb{P}^1_\Bbb{C}$ with limit set a single point, the case of $PSL(3,\Bbb{C})$ is far more subtle and intriguing. We show that there are five families of purely parabolic discrete groups in $PSL(3,\Bbb{C})$, and some of these actually split into subfamilies. We classify all these by means of their limit set and the control group. We use first the Lie-Kolchin Theorem and Borel's fixed point theorem to show that all purely parabolic discrete groups in $PSL(3,\Bbb{C})$ are virtually triangularizable. Then we prove that purely parabolic groups in $PSL(3,\Bbb{C})$ are virtually solvable and polycyclic, hence finitely presented. We then prove a slight generalization of the Lie-Kolchin Theorem for these groups: they are either virtually unipotent or else Abelian of rank 2 and of a very special type. All the virtually unipotent ones turn out to be conjugate to subgroups of the Heisenberg group ${\rm Heis}(3,\Bbb{C})$. We classify these using the obstructor dimension introduced by Bestvina, Kapovich and Kleiner. We find that their Kulkarni limit set is either a projective line, a cone of lines with base a circle or else the whole $\Bbb{P}^2_\Bbb{C}$. We determine the relation with the Conze-Guivarc'h limit set of the action on the dual projective space $\check{\Bbb{P}}^2_\Bbb{C}$ and we show that in all cases the Kulkarni region of discontinuity is the largest open set where the group acts properly discontinuously.

math.DS↗
arXiv2015

Two dimensional Veronese groups with an invariant ball

In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in $\Bbb{P}^2_{\Bbb{C}}$. As a corollary we get that every discrete compact surface group in $\PO^+(2,1)$ admits a deformation in $\PSL(3,\Bbb{C})$ 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 $\Bbb{P}^2_\Bbb{C}$.

math.DS↗
arXiv2015

The limit set for discrete complex hyperbolic groups

Given a discrete subgroup $\Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $\Bbb{H}^n_{\Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $\Gamma$ on all of $ \Bbb{P}^n_{\Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously and can be described as the complement of the union of all complex projective hyperplanes in $ \Bbb{P}^n_{\Bbb{C}}$ which are tangent to $\partial \Bbb{H}^n_{\Bbb{C}}$ at points in the Chen-Greenberg limit set $\Lambda_{CG}(\Gamma)$.

math.DS↗
arXiv2012

On Discrete Subgroups of automorphism of $P^2_C$

We study the geometry and dynamics of discrete subgroups $Γ$ of $\PSL(3,\mathbb{C})$ with an open invariant set $Ω\subset \PC^2$ where the action is properly discontinuous and the quotient $Ω/Γ$ contains a connected component whicis compact. We call such groups {\it quasi-cocompact}. In this case $Ω/Γ$ is a compact complex projective orbifold and $Ω$ is a {\it 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.

math.DS↗
arXiv2011

Projective Cyclic Groups in Higher Dimensions

In this article we provide a classification of the projective transformations in $PSL(n+1,\Bbb{C})$ considered as automorphisms of the complex projective space $\Bbb{P}^n$. Our classification is an interplay between algebra and dynamics, which just as in the case of isometries of CAT(0)-spaces, can be given by means of tree three types, namely: elliptic, parabolic and loxodromic. We carefully describe the dynamic in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, the discontinuity region and in some cases we provide families of maximal regions where the respective cyclic group acts properly discontinuously. Also we provide, in each case, some equivalents ways to classify the projective transformations.

math.DS↗
arXiv2010

The limit set of discrete subgroups of $PSL(3,\C)$

If $Γ$ is a discrete subgroup of $PSL(3,\Bbb{C})$, it is determined the equicontinuity region $Eq(Γ)$ of the natural action of $Γ$ on $\Bbb{P}^2_\Bbb{C}$. It is also proved that the action restricted to $Eq(Γ)$ is discontinuous, and $Eq(Γ)$ agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of $Γ$ in the sense of Kulkarni, $Λ(Γ)$, contains at least three lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which $Γ$ acts discontinuously. Moreover, if $Λ(Γ)$ contains at least four complex lines and $Γ$ acts on $\Bbb{P}^2_\Bbb{C}$ without fixed points nor invariant lines, then each connected component of $Eq(Γ)$ is a holomorphy domain and a complete Kobayashi hyperbolic space.

math.DG↗
arXiv2010

Subroups of $PSL(3,\Bbb{C})$ with four lines in general position in its Limit Set

In this article we provide an algebraic characterization of those groups of $PSL(3,\Bbb{C})$ 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.

math.DS↗
arXiv2008

On the Equicontinuity Region of Discrete Subgroups of PU(1,n)

Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $\mathbb {P}^n_\mathbb C$ preserving the unit ball $\mathbb {H}^n_\mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $\mathbb P^n_{\mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $\mathbb {P}^n_{\mathbb C}$ which are tangent to $\partial \mathbb {H}^n_\mathbb {C}$ at points in the Chen-Greenberg limit set $Λ_{CG}(G )$, a closed $G$-invariant subset of $\partial \mathbb {H}^n_\mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.

math.DS↗