Related papers: Test map and Discreteness in SL(2, $\mathbb H$)

The discreteness problem, that is, the problem of determining whether or not a given finitely generated group G of orientation preserving isometries of hyperbolic three-space is discrete as a subgroup of the whole isometry group of…

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…

We deal with two-generator subgroups of PSL(2,C) with real traces of both generators and their commutator. We give discreteness criteria for these groups when at least one of the generators is parabolic. We also present a list of the…

This paper obtains the matrix elements and characters of the discrete series unitary irreducible representations (UIRs) of the Sp$(4,\mathbb{R})$ group. With an isomorphic relationship to the two-fold covering of SO$_0(2,3)$…

For odd $n$ we construct a path $\rho_t\colon \pi_1(S) \to SL(n,\mathbb{R})$ of discrete, faithful and Zariski dense representations of a surface group such that $\rho_t(\pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$.

We provide the first examples of strongly dense representations of a hyperbolic 3-manifold group into $SL(4,\mathbb{R})$ and $SU(3,1)$ i.e. representations where every pair of non-commuting elements has Zariski dense image. Our examples are…

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature.…

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong…

Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic…

We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is {\it Zariski dense} in ${\rm PSL}(2,\,{\Bbb R})$. By using meromorphic differentials and affine connections, we obtain…

We characterize the maximal discrete subgroups of $SO^+(2,n+2)$, which contain the discriminant kernel of an even lattice, which contains two hyperbolic planes over $\mathbb{Z}$. They coincide with the normalizers in $SO^+(2,n+2)$ and are…

We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal…

For each integer $n \geq 3$, we exhibit a nonuniform arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.

We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an…

We study the dynamics of $SL_3(\mathbb{R})$ and its subgroups on the homogeneous space $X$ consisting of homothety classes of rank-2 discrete subgroups of $\mathbb{R}^3$. We focus on the case where the acting group is Zariski dense in…

We prove that all lattices of Sp(2n,R), except those commensurable with Sp(4k+2,Z) when n=2k+1, contain the image of infinitely many mapping class group orbits of Zariski-dense maximal representation that are continuous deformations of…

We explore the thinness of hypergeometric groups of type $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$ by applying a new approach of computer-assisted ping pong. We prove the thinness of $17$ hypergeometric groups with maximally unipotent monodromy…

Let $n, m\ge 2$. Let $\Gamma<\text{SO}^\circ(n+1,1)$ be a Zariski dense convex cocompact subgroup and $\Lambda\subset\mathbb{S}^n$ be its limit set. Let $\rho : \Gamma \to \text{SO}^\circ(m+1,1)$ be a Zariski dense convex cocompact faithful…

For any pair of orientable closed hyperbolic $3$--manifolds, this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense…

Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\phi:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside…