Related papers: A 2-dimensional Complex Kleinian Group With Infini…

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…

Given a discret subgroup $\Gamma\subset PSL(3,\C)$, we determine the number of complex lines and complex lines in general position lying in the complement of: maximal regions on which $\Gamma$ acts properly discontinuously, the Kularni's…

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…

We give a topological description of the quotient space $\Omega(G)/G$ in the case $G \subset PSL(3, \mathbb{C})$ is a discrete subgroup acting on $\mathbb{P}^2_\mathbb{C}$ and the maximum number of complex projective lines in general…

We compare different notions of limit sets for the action of Kleinian groups on the $n-$dimensional projective space via the irreducible representation $\varrho:PSL(2,\mathbb{C})\to PSL(n+1,\mathbb{C}).$ In particular, we prove that if the…

We study three problems related to the limit sets of discrete subgroups of PSL(n+1,C). In Chapter 2, we study the dynamics of solvable discrete subgroups of PSL(n+1,C). We prove that solvable groups are virtually triangularizable and we…

If $\Gamma$ is a discrete subgroup of $PSL(3,\Bbb{C})$, it is determined the equicontinuity region $Eq(\Gamma)$ of the natural action of $\Gamma$ on $\Bbb{P}^2_\Bbb{C}$. It is also proved that the action restricted to $Eq(\Gamma)$ is…

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…

We study the geometry and dynamics of discrete subgroups $\Gamma$ of $\PSL(3,\mathbb{C})$ with an open invariant set $\Omega \subset \PC^2$ where the action is properly discontinuous and the quotient $\Omega/\Gamma$ contains a connected…

Classical Kleinian groups are discrete subgroups of $PSL(2,\C)$ acting on the complex projective line $\P^1$, which actually coincides with the Riemann sphere, with non-empty region of discontinuity. These can also be regarded as the…

In this note, we show that the exceptional algebraic set of an infinite discrete group in $PSL(3,\Bbb{C})$ should be a finite union of complex lines, copies of the Veronese curve or copies of the cubic $xy^2-z^3$.

In this article, we show that there exist discrete isometry groups of the $2$- and $3$-dimensional complex hyperbolic spaces with critical exponents arbitrarily close to but strictly smaller than the maximum possible value. This result…

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…

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…

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group…

We study finite groups arising from configurations of pairwise skew lines in $\mathbb{P}^3_K$. To such a configuration ${L}$ one associates a group $G_{L}\subset \mathrm{PGL}_2(K)$ acting on each line, and we investigate which finite…

For a torsion free Kleinian group $\Gamma$ without parabolics, we consider the decomposition of the limit set $L(\Gamma)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(\Gamma)$…

In this article, we prove a version of Martin and Skora's conjecture that convergence groups on the $2$-sphere are covered by Kleinian groups. Given a relatively hyperbolic group pair $(G,\mathcal{P})$ with planar boundary and no Sierpinski…

We shall explain here an idea to generalize classical complex analytic Kleinian group theory to any odd dimensional cases. For a certain class of discrete subgroups of $\PGL_{2n+1}(\C)$ acting on $\P^{2n+1}$, we can define their domains of…

While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider subgroups…