arXiv research

Surveying Hitchin representations of Fuchsian groups.

Generic Hitchin representations generate dense subgroups.

New findings on cusped Borel Anosov representations and their properties.

Generic Hitchin representations avoid hyperplanes in Lie algebras.

Study pressure metrics for cusped Hitchin representations.

The abstract discusses rational approximations for Hitchin representations on surfaces.

In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy …

Entropy rigidity theorem for cusped Hitchin representations.

Characterizes flag geometries for Hitchin representations in SL3(R).

In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the u…

Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.

In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.

Develops correlation number for specific potentials and Hitchin representations.

We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in t…

We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.

Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$ are proven to be Hitchin.

Study on projective structures linked to Hitchin representations.

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

Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the "length" …

New insights into currents of Hitchin representations with combinatorial restrictions.

Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.

Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).

We prove an extension of Basmajian's identity to $n$-Hitchin representations of compact bordered surfaces. For $n=3$, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the L…

Strict plurisubharmonicity proven for Teichmüller energy on Hitchin representations.

The paper finds dense subgroups in certain Lie groups.

In this paper we show some properties of triangle invariants and shearing invariants of PSL(n,R)-Fuchsian representations. Moreover, using the Bonahon-Dreyer parameterization, we show that the Fuchsian locus of Hitchin components corresponds to a slice.

In this paper, we study the monodromy of the Hitchin fibration for rank 2 vector bundles over hyperelliptic curves. We reduce the problem to studying a surface braid group generalization of the classical Burau representation, and give a combinatorial method for computing this representation.

New metrics for Anosov representations defined from Thurston's asymmetric metrics.

The study describes the geometry of surfaces and their representations in SL(3,R).

Compactifies a component by studying metric degeneration.

Cataclysm deformations study Anosov representations and their convergence.

Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.

Study k-positive surface group representations and their degenerations.

The Hitchin component is a connected component of the character variety of reductive group homomorphisms from the fundamental group of a closed surface S of genus greater than 1 to the Lie group PSL_m(R). The Teichmuller space of S naturally embeds into the Hitchin component. The limit points in the Thurston compactifi…

Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\eq…

We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies Hitchin's moduli space with the moduli space of flat $K^\mathbb{C}$-connections,…

New compactification for character varieties with good topological properties.

This paper connects real closed fields to Hitchin representations and their properties.

This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic …

Geometric methods for surface group representations in higher rank SL(2m+1,R).

Collar lemma proven for certain surface group representations.

New metrics derived from Hölder distortion on Hitchin components.

The paper generalizes deformation results for Fuchsian representations and shows proper affine actions.

Classifies Zariski closures of positive representations in Lie groups.

In this article we use the "escape from subvarieties lemma" introduced by Eskin--Mozes--Oh to prove finite step rigidity results for the Jordan-Lyapunov projection spectra of Hitchin representations and the Margulis-Smilga invariant spectra of some special Margulis-Smilga spacetimes. In the process, we also prove a sim…

We consider the moduli space of polystable $L$-twisted $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a real reductive Lie group, and $L$ is a holomorphic line bundle over $X$. Evaluating the Higgs field at a basis of the ring of polynomial invariants of the isotropy representation, one defines the …

Using Hitchin's parameterization of the Hitchin-Teichmüller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solvin…

Properly discontinuous actions of a surface group by affine automorphisms of $\mathbb R^d$ were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that o…