arXiv research

New lattices in higher dimensions have dense surface subgroups.

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

The study finds conditions for certain groups to be dense in a specific mathematical space.

Generic Hitchin representations generate dense subgroups.

The paper finds dense subgroups in certain Lie groups.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

Odd-dimensional SL(n,Q) contains dense surface subgroups.

Proves unique maps from certain spaces to others.

Regular subgroups of SL3(R) are identified and ruled out.

The study explores deformations of discrete subgroups in non-compact homogeneous spaces.

Paper finds surface groups can deform in reductive symmetric spaces.

Example shows dense subgroup of SL5(Z) not finitely presented.

New subgroup found in Lie groups with unusual properties.

Maximal representations in symplectic lattices proven for most cases.

Research examines coamenable subgroups in higher rank groups.

New representations of hyperbolic 3-manifold groups into larger groups.

The paper proves a unique conformal measure for Anosov groups and shows local mixing.

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 dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_…

Study thin hyperbolic reflection groups and their properties.

Study critical exponents in normal subgroups of higher rank Lie groups.

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev…

Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional $\mathbb F$-hyperbolic space. Let ${\rm U}(n,1; \mathbb F)$ be the linear group that acts by the isometries. A subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is called \emph{Zariski dense} if it does not fix a point…

The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.

Study subgroups preserving proper domains in flag manifolds.

New domains of discontinuity found for Anosov representations.

The study proposes a conjecture about the monodromy group of singular hyperbolic metrics and provides evidence and confirmations.

We show that every limit point of a Zariski dense discrete subgroup $Γ$ of the isometry group of a symmetric space of noncompact type is conical if and only if $Γ$ is convex cocompact.

Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.

Study shows mixing of flows on specific geometric spaces.

The study examines growth of quadratic forms under Anosov subgroups.

The study explores deformations of standard locally homogeneous spaces.

Deforms surface groups to be Zariski dense in SL(n,R)

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

We build examples of properly convex projective manifold $Ω/ Γ$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$ which are not lattice, nor Schottky groups. Moreover, the open properly convex set $Ω$ is…

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak …

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

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as…

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 λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, t…

The paper bounds growth indicator functions for discrete subgroups in algebraic groups.

We present a new proof of the following theorem of Benoist-Quint: Let $G:=SO^\circ(d,1)$, $d\ge 2$ and $Δ<G$ a cocompact lattice. Any orbit of a Zariski dense subgroup $Γ$ of $G$ is either finite or dense in $Δ\backslash G$. While Benoist and Quint's proof is based on the classification of stationary measures, our proo…

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense sub…

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of the five dimensional (real) hyperbolic space. We obtain discreteness criteria f…

Study shows how to detect representation extendability using conformal measures.

Study on horospheres in higher rank homogeneous spaces, proving density properties.

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on…

The study proves conditions for rigidity of Kleinian groups using measure theory and ergodic theory.

Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus $g\geq 3$ closed orientable surface at a prime $p\geq 5$ is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\Q$. As an applicat…