arXiv research

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.

Maximal representations in symplectic lattices proven for most cases.

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

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

New domains of discontinuity found for Anosov representations.

New lattices in higher dimensions have dense surface subgroups.

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

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

Study shows how to detect representation extendability using conformal measures.

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.

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…

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…

New subgroup found in Lie groups with unusual properties.

Proves unique maps from certain spaces to others.

Hyperbolic groups' infinite orbits spread evenly in spaces.

We study the TQFT mapping class group representations for surfaces with boundary associated with the $SU(2)$ gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental valu…

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.

The study examines growth of quadratic forms under Anosov subgroups.

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

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

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…

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

Anosov representations of surface groups are complex manifolds.

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…

Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.

In this note, we study deformations of discrete and Zariski dense subgroups of SU(2, 1) in quaternionic hyperbolic space. Specifi- cally we consider two examples coming from representations of 3-manifold groups (the figure eight knot and Whitehead links complement) and show opposite behavior: one is not deformable outs…

New examples of embeddings defy Anosov representation limits.

Classifies Zariski closures of positive representations in Lie groups.

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

The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.

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.

Research examines coamenable subgroups in higher rank groups.

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

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}_…

Study trisections on rational elliptic surfaces to find new Zariski pairs.

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 establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exce…