arXiv subjects

Giuseppe Martone

Publications and source records attributed to Giuseppe Martone.

11 recordsLinked to original sources

arXiv2026

Holmes-Thompson area of inscribed polygons and convex projective structures

Positive tuples of complete flags in $\mathbb{R}^3$ define two convex polygons in $\mathbb{RP}^2$, one inscribed in the other. We are interested in relating the Holmes-Thompson area of the inner polygon for the Hilbert metric on the outer polygon to the double and triple ratios of the positive tuple of flags. This article focuses on positive triples and quadruples of flags. For quadruples, we investigate the special cases of hyperbolic quadrilaterals and the parametrization of the finite area convex real projective structures on a thrice-punctured sphere.

math.GT↗

arXiv2025

Topology of the space of $d$-pleated surfaces

Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained by applying generalized bending along $\lambda$ to Hitchin representations. When $d=2$, one recovers abstract pleated surfaces in $\mathbb{H}^3$. In this paper, we study the topology of the space $\mathfrak{R}(\lambda,d)$ of conjugacy classes of $d$-pleated surfaces with pleating locus $\lambda$. Firstly, we prove that $\mathfrak{R}(\lambda,d)$ is real-analytically diffeomorphic to $\mathbb{R}^{(d^2-1)(2g-2)}\times(\mathbb{R}/2\pi\mathbb{Z})^{(d^2-1)(2g-2)}\times \mathbb{Z}_d$, where $\mathbb{Z}_d$ denotes the finite cyclic group of order $d$. Furthermore, we show that each connected component of the space of conjugacy classes in $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ contains exactly one component of $\mathfrak{R}(\lambda,d)$.

math.GT↗

arXiv2024

Correlation Number for Potentials with Entropy Gaps and Cusped Hitchin Representations

We introduce a correlation number for two strictly positive, locally H\"older continuous, independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with BIP. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.

math.DS↗

arXiv2023

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 into PSL(d,R). We produce pressure metrics associated to the first fundamental weight and the first simple root. When $d=3$ we produce a pressure metric associated to the Hilbert length, which is new even when the Fuchsian group is cocompact.

math.GT↗

arXiv2023

$d$-pleated surfaces and their shear-bend coordinates

In this article, we single out representations of surface groups into $\mathsf{PSL}_d(\mathbb{C})$ which generalize the well-studied family of pleated surfaces into $\mathsf{PSL}_2(\mathbb{C})$. Our representations arise as sufficiently generic $\lambda$-Borel Anosov representations, which are representations that are Borel Anosov with respect to a maximal geodesic lamination $\lambda$. For fixed $\lambda$ and $d$, we provide a holomorphic parametrization of the space $\mathcal{R}(\lambda,d)$ of $(\lambda,d)$-pleated surfaces which extends both work of Bonahon for pleated surfaces and Bonahon and Dreyer for Hitchin representations.

math.GT↗

arXiv2022

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

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.

math.DS↗

arXiv2021

Correlation of the renormalized Hilbert length for convex projective surfaces

In 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.

math.GT↗

arXiv2021

A closed ball compactification of a maximal component via cores of trees

We show that, in the character variety of surface group representations into the Lie group $\mathrm{PSL}(2,\mathbb{R}) \times \mathrm{PSL}(2,\mathbb{R})$, the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts. We study the dynamics of this action. Finally, we describe the boundary points geometrically as $(\overline{A_{1} \times A_{1}},2)$-valued mixed structures.

math.GT↗

arXiv2019

Sequences of Hitchin representations of Tree-type

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 compactification of the Teichmuller space are well-understood. Our main goal is to provide non-trivial sufficient conditions on a sequence of Hitchin representations so that the limit of this sequence in the Parreau boundary can be described as an action on a tree. These non-trivial conditions are given in terms of Fock-Goncharov coordinates on moduli spaces of generic tuples of flags.

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arXiv2019

Positively ratioed representations

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.

math.GT↗

arXiv2019

Positive configurations of flags in a building and limits of positive representations

Parreau compactified the Hitchin component of a closed surface $S$ of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of $π_1(S)$ on an $\mathbb R$-Euclidean building. In this paper, we use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building.

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