Related papers: Schottky groups cannot act on $\mathbb{P}^{2n}_{\m…
We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups,…
The goal of this article is to initiate the study of estimates of the non-classical Schottky structure in the discrete subgroups of the projective special linear group over the real numbers degree $2$. In fact, in this paper, we have…
In this article we present an example of a discrete group $\Sigma_\C\subset PSL(3,\Bbb{R})$ whose action on $\P^2$ does no have invariant projective subspaces, is not conjugated to complex hyperbolic group and its limit set in the sense of…
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 show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSL_n(q) and PSp_{2n}(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one…
We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly…
We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb…
We systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori's well-known construction. This yields new examples of non-K\"ahler compact complex manifolds having free…
We use the classical construction of Schottky groups in hyperbolic geometry to produce non-Schottky subgroups of the mapping class group.
A virtual Schottky group is a Kleinian group $K$ containing a Schottky group $G$ as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of…
The theoretical existence of non-classical Schottky groups is due to Marden. Explicit examples of such kind of groups are only known in rank two, the first one by by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the…
In this work we study and provide a full description, up to a finite index subgroup, of the dynamics of solvable complex Kleinian subgroups of PSL(3,C). These groups havesimpledynamics, contrary to strongly irre-ducible groups. Because of…
The principal character of a representation of the free group of rank two into PSL(2, C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of…
Schottky space ${\mathcal S}_{g}$, where $g \geq 2$ is an integer, is a connected complex orbifold of dimension $3(g-1)$; it provides a parametrization of the ${\rm PSL}_{2}({\mathbb C})$-conjugacy classes of Schottky groups $\Gamma$ of…
We prove that PSL(2,Z[1/p]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL(2,Z[1/p]) and PSL(2,Z[1/q]) are not quasi-isometric unless p=q, and, independent…
We investigate for which linear-algebraic groups (over the complex numbers or any local field) there exists subgroups which are dense in the Zariski topology, but discrete in the Hausdorff topology. For instance, such subgroups exist for…
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 consider the natural action of the quaternionic projective linear group $\mathrm{PSL}(n+1,\mathbb{H})$ on the quaternionic projective space $\mathbb{P}^n_{\mathbb{H}}$. We compute the Kulkarni limit sets for the cyclic subgroups of…
Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity…
In this paper we provide the complete classification of convex cocompact Kleinian group of Hausdorff dimensions less than $1.$ In particular, we prove that every convex cocompact Kleinian group of Hausdorff dimension $<1$ is a classical…