arXiv research

New lattices in higher dimensions have dense surface subgroups.

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

Generic Hitchin representations generate dense subgroups.

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

The paper finds dense subgroups in certain Lie groups.

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

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic 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…

Maximal representations in symplectic lattices proven for most cases.

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

Proves unique maps from certain spaces to others.

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

Hyperbolic groups' infinite orbits spread evenly in spaces.

Complex projective manifolds without rational curves are quotients of Abelian varieties.

Paper finds surface groups can deform in reductive symmetric spaces.

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.

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

Integral points are potentially dense in character varieties of quasi-projective varieties.

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

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

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…

New domains of discontinuity found for Anosov representations.

Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.

New subgroup found in Lie groups with unusual properties.

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

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.

The paper explores mapping class group quotients by Dehn twists and their representations.

Research examines coamenable subgroups in higher rank groups.

Study shows how to detect representation extendability using conformal measures.

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

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.

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}_…

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

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…

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…

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.

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existenc…

Study thin hyperbolic reflection groups and their properties.

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

The study explores deformations of standard locally homogeneous spaces.

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…

Counting spheres in hyperbolic space with effective methods.

Study shows mixing of flows on specific geometric spaces.

The study examines growth of quadratic forms under Anosov subgroups.

Study subgroups preserving proper domains in flag manifolds.

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…

Limit sets of $\mathrm{AdS}$-quasi-Fuchsian groups of $\mathrm{PO}(n,2)$ are always Lipschitz submanifolds. The aim of this article is to show that they are never $\mathcal{C}^1$, except for the case of Fuchsian groups. As a byproduct we show that $\mathrm{AdS}$-quasi-Fuchsian groups that are not Fuchsian are Zariski d…