arXiv research

Classifies Zariski closures of positive representations in Lie groups.

Study of translation covers of platonic solids reveals monodromy group structures.

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.

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

The paper studies representations of braid groups via curves and finds conditions for their Zariski closure and arithmeticity.

In this paper we aim at the description of foliations having tangent sheaf $T\mathcal F$ with $c_1(T\mathcal F)=c_2(T\mathcal F)=0$ on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of $\mathcal F$ is an…

The study explores deformations of standard locally homogeneous spaces.

The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.

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

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski to…

We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.

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

Study Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.

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 introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

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…

We consider a certain hybridization construction which produces a subgroup of ${\rm PU}(n,1)$ from a pair of lattices in ${\rm PU}(n-1,1)$. Among the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$, we show that the hybrid of pairs of Fuchsian subgroups ${\rm PU}(1,1,\mathcal{O}_d)$ is a lattice when $d=1$ and $d=7…

The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit …

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are d…

The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starti…

A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically fi…

Holomorphic tensors on algebraic cones are invariant under certain group actions.

Study the expressivity and training complexity of polynomial neural networks.

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $Γ\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/Γ\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $Γ$ in $\Iso(\RR^n_s)$. Moreover, we show that the $G$-orbits in $\RR^n_s$ are affinely diffeomorphic …

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non…

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

Using the thermodynamics formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce a $Out(Γ)$-invariant Riemannian metric on the smooth …

New lattices in higher dimensions have dense surface subgroups.

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.

Two unique conic-line arrangements with degree 9 are found.

The paper finds dense subgroups in certain Lie groups.

Study conic line arrangements of degree 7, finding their topology and connected components.

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…

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…

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

Generic Hitchin representations generate dense subgroups.

Maximal representations in symplectic lattices proven for most cases.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

Proves unique maps from certain spaces to others.

Character variety of Whitehead link described in detail.

We study the character variety of representations of the fundamental group of a closed surface of genus $g\geq2$ into the Lie group SO(n,n+1) using Higgs bundles. For each integer $0<d\leq n(2g-2),$ we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain…

Hyperbolic groups' infinite orbits spread evenly in spaces.