arXiv subjects

León Carvajales

Publications and source records attributed to León Carvajales.

6 recordsLinked to original sources

arXiv2026

The triangular metric on projective surfaces

In \cite{ThurstonStretch} Thurston introduced stretch paths and stretch maps between hyperbolic surfaces. In this paper we extend similar constructions to the space of convex real projective structures on surfaces. Using Fock-Goncharov-like parameters, we introduce stretch paths and construct explicit stretch maps between the projective surface. We show that stretch maps are $e^t$--Lipschitz with respect to a triangular metric we introduce on the projective surface. This allows us to gain control on several length functions on an open set of the space of convex real projective structures (those where there is no ``local back--tracking"). As an application we obtain pointwise upper bounds for the Hilbert entropy of some convex real projective structures in terms of Fock-Goncharov-like parameters.

math.DG↗

arXiv2024

Anosov representations acting on homogeneous spaces: domains of discontinuity

We construct open domains of discontinuity for Anosov representations acting on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces. This generalizes work of Kapovich-Leeb-Porti on flag spaces. Our results complement those of Guéritaud-Guichard-Kassel-Wienhard, who constructed proper actions of Anosov representations. For Zariski dense Anosov representations with respect to a minimal parabolic subgroup acting on some symmetric spaces, we show that our construction describes the largest possible open domains of discontinuity.

math.GT↗

arXiv2024

Quasi-isometric free group representations into SL_3(R)

We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

math.GR↗

arXiv2024

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.

math.DG↗

arXiv2021

Growth of quadratic forms under Anosov subgroups

Let $ρ:Γ\rightarrow PSL_d(\mathbb{K})$ be a Zariski dense Borel-Anosov representation, for $\mathbb{K}$ equal to $\mathbb{R}$ or $\mathbb{C}$. Let $o$ be a form of signature $(p,d-p)$ on $\mathbb{K}^d$ (where $0<p<d)$. Let $\mathsf{S}^o$ be the corresponding geodesic copy of the Riemannian symmetric space of $PSO(o)$, inside the Riemannian symmetric space of $PSL_d(\mathbb{K})$. For certain choices of $o$ and every $t$ large enough, we show exponential bounds for the number of $γ\inΓ$ for which the distance between $\mathsf{S}^o$ and $ργ\cdot\mathsf{S}^o$ is smaller than $t$. Under an extra assumption, satisfied for instance when the boundary of $Γ$ is connected, we show an asymptotic as $t\rightarrow\infty$ for the counting function relative to a functional in the interior of the dual limit cone.

math.GR↗

arXiv2019

Counting problems for special-orthogonal Anosov representations

For positive integers $p$ and $q$ let $G:=\textrm{PSO}(p,q)$ be the projective indefinite special-orthogonal group of signature $(p,q)$. We study counting problems in the Riemannian symmetric space $X_G$ of $G$ and in the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q-1}$. Let $S\subset X_G$ be a totally geodesic copy of $X_{\textrm{PSO}(p,q-1)}$. We look at the orbit of $S$ under the action of a projective Anosov subgroup of $G$. For certain choices of such a geodesic copy we show that the number of points in this orbit which are at distance at most $t$ from $S$ is finite and asymptotic to a purely exponential function as $t$ goes to infinity. We provide an interpretation of this result in $\mathbb{H}^{p,q-1}$, as the asymptotics of the amount of space-like geodesic segments of maximum length $t$ in the orbit of a point.

math.GR↗