Related papers: A characterization of complex hyperbolic Kleinian …

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 $\Gamma$ 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.

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 $\mathrm{U}(2,1)$.

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 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 $M = H^3/\Gamma$ be a hyperbolic 3-manifold, where $\Gamma$ is a non-elementary Kleinian group. It is shown that the length spectrum of $M$ is of unbounded multiplicity.

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…

Let $A_\Gamma$ be an Artin group with defining graph $\Gamma$. We introduce the notion of $A_\Gamma$ being extra-large relative to a family of arbitrary parabolic subgroups. This generalizes a related notion of $A_\Gamma$ being extra-large…

In this note, we study the field generated by the traces of subgroups of SU(n,1). Under some hypotheses, the trace field of a group $\Gamma \subset$ SU(2,1) is equal to the field generated by the coefficients of the matrices in $\Gamma$. If…

Let $A$ be a finite dimensional $Q-$algebra and $\Gamma subset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(\Gamma)$. We call this last property the hyperbolic property. We apply this in…

We consider non-elementary Kleinian groups \Gamma, without invariant plane, generated by an elliptic and a hyperbolic element with their axes lying in one plane. We find presentations and a complete list of orbifolds uniformized by such…

This paper investigates the cuspidal spectrum of the quotient of the real Lie group $G= SU(n,1)$ and a principal congruence subgroup $\Gamma(m)$ for $m\geq 3$, focusing on the multiplicities of integrable discrete series representations.…

Let $\Gamma$ be a non-elementary, non-convex-cocompact Kleinian group acting on $\mathbb{H}^{d}$. We show that the Hausdorff dimension of the sublinearly conical Myrberg limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$.…

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.

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…

We show that $S^n \vee S^m$ is $\mathbb{Z}/p^r$-hyperbolic for all primes $p$ and all $r \in \mathbb{Z}^+$, provided $n,m \geq 2$, and consequently that various spaces containing $S^n \vee S^m$ as a $p$-local retract are…

The compact hyperbolic triangle group $\Delta(p,q,r)$ admits a canonical representation to $\mathrm{PSL}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K =…

We give a short new proof that for each non-elementary Kleinian group $\Gamma$, the exponent of convergence of an arbitrary non-trivial normal subgroup is bounded below by half of the exponent of convergence of $\Gamma$, and that strict…

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 $N(\Gamma,G)$ be the number of homomorphisms from $\Gamma$ to $G$ up to conjugation by $G$. Physics of four-dimensional $\mathcal{N}=4$ supersymmetric gauge theories predicts that $N(\Gamma,G)=N(\Gamma , \tilde G)$ when $\Gamma$ is a…