Hitchin Representations Research Articles

Degenerations of k$k$‐positive surface group representations

AbstractWe introduce ‐positive representations, a large class of ‐Anosov surface group representations into that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non‐discrete representations, but any limit is at least ‐positive and irreducible limits are ‐positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations.

Planar minimal surfaces with polynomial growth in the Sp(4,ℝ)-symmetric space

abstract: We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\Sp(4,\R)$-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild $\Sp(4,\R)$-Higgs bundles over $\CP^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\h^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\R^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\h^{2,2}$ associated to $\Sp(4,\R)$-Hitchin representations along rays of holomorphic quartic differentials.

Thurston’s asymmetric metrics for Anosov representations

We provide a good dynamical framework allowing to generalize Thurston’s asymmetric metric and the associated Finsler norm from Teichmüller space to large classes of Anosov representations. In many cases, including the space of Hitchin representations, this gives a (possibly asymmetric) Finsler distance. In some cases, we explicitly compute the associated Finsler norm.

Gap Between Lyapunov Exponents for Hitchin Representations

Abstract We study Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We consider them as invariants on the space of Hitchin representations and show that there is a gap between any two consecutive Lyapunov exponents. Moreover we show that the minimal possible gap for any two consecutive Lyapunov exponents is achieved if and only if the representation is the one uniformizing the hyperbolic structure of the surface. We give two proofs of the previous fact. In the first one, we relate the Lyapunov exponents to a transverse Lyapunov exponent associated to a deformation of the unstable foliation of the geodesic flow, and we establish a general bound for this quantity. In the second one, we relate Lyapunov exponents to the renormalized intersection product in the setting of the thermodynamic formalism developed by Bridgeman, Canary, Labourie, and Sambarino and we use the already existing bound for such a quantity.

The Covariance Metric in the Blaschke Locus

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichmüller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in Guillarmou et al. in (Ergod Theory Dyn Syst 43:974–1022, 2021). We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.

Cusped Borel Anosov representations with positivity

We show that if a cusped Borel Anosov representation from a lattice Γ⊂PGL2(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\subset \ extsf{PGL}_2({{\\,\\mathrm{\\mathbb {R}}\\,}})$$\\end{document} to PGLd(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{PGL}_d({{\\,\\mathrm{\\mathbb {R}}\\,}})$$\\end{document} contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.

Most Hitchin Representations Are Strongly Dense

Most Hitchin Representations Are Strongly Dense

Pressure metrics for cusped Hitchin components

We study the cusped Hitchin component consisting of (conjugacy classes of) cusped Hitchin representations of a torsion-free geometrically finite Fuchsian group Γ into PSL(d,R). We produce Riemannian pressure metrics associated to the first fundamental weight and the first simple root. We produce a pressure path metric associated to the Hilbert length and describe its degeneracy.

Hitchin representations of Fuchsian groups

We survey the theory of Hitchin representations of closed surface groups into \mathsf{PSL}(d,\mathbb R) with a focus on their dynamical and geometric properties. We then describe recent extensions of this work to study Hitchin representations of co-finite area Fuchsian groups. The motivation for this recent work is a conjecture about the geometry of the augmented Hitchin component.

Correlation of the renormalized Hilbert length for convex projective surfaces

AbstractIn this paper, we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The correlation number in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of Schwartz and Sharp. In contrast, we provide examples of diverging sequences, defined via cubic rays, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.

Cataclysms for Anosov representations

In this paper, we construct cataclysm deformations for theta -Anosov representations into a semisimple non-compact connected real Lie group G with finite center, where theta subset Delta is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston’s cataclysms on Teichmüller space and Dreyer’s cataclysms for Borel-Anosov representations into mathrm {PSL}(n, mathbb {R}). We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for theta -Anosov representations.

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

Abstract We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu.

Domains of discontinuity in oriented flag manifolds

We study actions of discrete subgroups Γ $\Gamma$ of semi-simple Lie groups G $G$ on associated oriented flag manifolds. These are quotients G / P $G/P$ , where the subgroup P $P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups Γ ⊂ G $\Gamma \subset G$ , we identify domains in oriented flag manifolds by removing a set obtained from the limit set of Γ $\Gamma$ , and give a combinatorial description of proper discontinuity and cocompactness of these domains. This generalizes analogous results of Kapovich–Leeb–Porti to the oriented setting. We give first examples of cocompact domains of discontinuity which are not lifts of domains in unoriented flag manifolds. These include in particular domains in oriented Grassmannians for Hitchin representations, which we also show to be nonempty. As a further application of the oriented setup, we give a new lower bound on the number of connected components of B $B$ -Anosov representations of a closed surface group into SL ( n , R ) $\operatorname{SL}(n,\mathbb {R})$ .

Hessian of Hausdorff dimension on purely imaginary directions

Bridgeman–Taylor (Math. Ann. 341 (2008), 927–943) and McMullen (Invent. Math. 173 (2008), 365–425) showed that the Weil–Petersson metric on Teichmüller space can be realized by looking at the infinitesimal change of the Hausdorff dimension of certain quasi-Fuchsian deformations. In this article, we give a similar geometric interpretation of the spectral gap pressure metric introduced by Bridgeman–Canary–Labourie–Sambarino (Geom. Dedicata 192 (2018), 57–86) on the Hitchin component for PSL d ( R ) ${{\mathsf {PSL}}}_d(\mathbb {R})$ . More generally, we investigate the Hessian of the Hausdorff dimension as a function on the space of (1,1,2)-hyperconvex representations, a class introduced in (J. reine angew. Math. 774 (2021), 1–51) which includes small complex deformations of Hitchin representations and of Θ $\Theta$ -positive representations. As another application, we prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ → PO ( n , 1 ) → PU ( n , 1 ) ${{\Gamma }}\rightarrow {{\mathsf {PO}}}(n,1)\rightarrow {{\mathsf {PU}}}(n,1)$ is positive definite when Γ ${{\Gamma }}$ is co-compact in PO ( n , 1 ) ${{\mathsf {PO}}}(n,1)$ (unless n = 2 $n=2$ and the deformation is tangent to X ( Γ , PO ( 2 , 1 ) ) $\mathfrak {X}({{\Gamma }}, {{\mathsf {PO}}}(2,1))$ ).

Cross ratios and cubulations of hyperbolic groups

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary partial _{infty }G. A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a mathrm{CAT(0)} cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Projective Anosov representations, convex cocompact actions, and rigidity

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain. We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a $C^2$ submanifold of projective space. This final rigidity result also applies to Hitchin representations.

Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations

Let Σ \Sigma be a closed surface of genus at least two and ρ : π 1 ( Σ ) → G \rho \colon \pi _1(\Sigma ) \to G a Hitchin representation into G = P S L ( n , R ) G=PSL(n,\mathbb {R}) , P S p ( 2 n , R ) PSp(2n,\mathbb {R}) , P S O ( n , n + 1 ) PSO(n,n+1) or G 2 G_2 . We consider the energy functional E E on the Teichmüller space of Σ \Sigma which assigns to each point in T ( Σ ) \mathcal {T}(\Sigma ) the energy of the associated ρ \rho -equivariant harmonic map. The main result of this paper is that E E is strictly plurisubharmonic. As a corollary we obtain an upper bound of 3 ⋅ genus ( Σ ) − 3 3 \cdot \text {genus}(\Sigma ) -3 on the index of any critical point of the energy functional.

A collar lemma for partially hyperconvex surface group representations

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into S L ( n , R ) SL(n,\mathbb {R}) that satisfy partial hyperconvexity properties inspired from Labourie’s work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ \Theta -positive representations into S O ( p , q ) SO(p,q) if p ≥ 4 p\geq 4 . We moreover show that ‘positivity properties’ known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.

The real spectrum compactification of character varieties: characterizations and applications

We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations.