Related papers: Complex and Quaternionic hyperbolic Kleinian group…

Let $\Gamma$ be a nonelementary discrete subgroup of $\mathrm{Sp}(n,1)$. We show that if the trace skew-field of $\Gamma$ is commutative, then $\Gamma$ stabilizes a copy of complex hyperbolic subspace of quaternionic hyperbolic $n$-space.

We show that $\Gamma < \textbf{SU}(3,1)$ is a non-elementary complex hyperbolic Kleinian group in which $tr(\gamma) \in \R$ for all $\gamma \in \Gamma$ if and only if $\Gamma$ is conjugate to a subgroup of $\textbf{SO}(3,1)$ or…

We show that if $\Gamma$ is an irreducible subgroup of ${\rm SU}(2,1)$, then $\Gamma$ contains a loxodromic element $A$. If $A$ has eigenvalues $\lambda_1 = \lambda e^{i\varphi},$ $\lambda_2 = e^{-2i\varphi}$, $\lambda_3 =…

Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation…

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is…

Let $G$ be a non-elementary discrete subgroup of $\mathrm{Sp}(2,1)$. We show that if the sum of diagonal entries of each element of $G$ is a complex number, then $G$ is conjugate to a subgroup of $\mathrm{U}(2,1)$.

In this notes, we characterize discrete subgroups of PU(2,1), holomorphic isometric group of complex hyperbolic space, which have an invariant totally geodesic submanifold.

Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski…

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup $\Gamma$ of G, we establish equidistribution of holonomy classes about closed geodesics for the associated locally symmetric space. Our result is given in a…

Let $\Gamma \stackrel{i}{\hookrightarrow} L$ be a lattice in the real simple Lie group $L$. If $L$ is of rank at least 2 (respectively locally isomorphic to $Sp(n,1)$) any unbounded morphism $\rho: \Gamma \longrightarrow G$ into a simple…

We show that if $G$ is a real semi-simple Lie group, and $\Gamma$ is a discrete subgroup of $G$ containing a subgroup $\Sigma$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $\Gamma$ is not isomorphic to a…

Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number \lambda_0…

We construct the first example of a Zariski-dense, discrete, non-lattice subgroup $\Gamma_0$ of a higher rank simple Lie group $G$, which is non-tempered in the sense that the quasi-regular representation $L^2(\Gamma_0\backslash G)$ is…

Let $X$ be a negatively curved symmetric space and $\Gamma$ a non-cocompact lattice in $\rm{Isom}(X)$. We show that small, parabolic-preserving deformations of $\Gamma$ into the isometry group of any negatively curved symmetric space…

Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let \Gamma < G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of \Gamma (whose…

For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= \Gamma \backslash \mathbb{H}^{n}$ where $\Gamma$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential…

If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct…

Let $\Gamma$ be a non-elementary Kleinian group and $H<\Gamma$ a finitely generated, proper subgroup. We prove that if $\Gamma$ has finite co-volume, then the profinite completions of $H$ and $\Gamma$ are not isomorphic. If $H$ has finite…

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical…

Let $X=G/K$ be an irreducible Hermitian symmetric space of the non-compact type and let $S\in G^\mbb{C}$ be the associated compression semi-group. Let $\Gamma$ be a discrete subgroup of $G$. We give a sufficient condition for…