arXiv subjects

Andres Sambarino

Publications and source records attributed to Andres Sambarino.

5 recordsLinked to original sources

arXiv2026

Bending, entropy and proper affine actions of surface groups

We show that for any closed surface $S$ there is an explict neighborhood $V$ of the fuchsian locus in quasifuchsian space $\mathsf{QF}(S)$ such that for every representation $ρ\in V$ which is not fuchsian, there is a proper affine action on $\mathfrak{sl}(2,\mathbb{C})$ with linear part $\mathsf{Ad}(ρ)$. We further show that there is a larger neighborhood $U$ of the Fuchsian locus so that every critical point of the entropy function in $U$ lies on the Fuchsian locus.

math.GT↗

arXiv2017

Simple root flows for Hitchin representations

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.

math.DG↗

arXiv2015

The pressure metric for Anosov representations

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 points of the deformation space of irreducible, projective Anosov representations of a word hyperbolic group $Γ$ into $SL(m,R)$ whose Zariski closure contains a generic element. In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil--Petersson metric on the Fuchsian loci. Moreover, we produce $Out(Γ)$-invariant metrics on deformation spaces of convex cocompact representations into $PSL(2,C)$ and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

math.DG↗

arXiv2014

The orbital counting problem for hyperconvex representations

We give a precise counting result on the symmetric space of a noncompact real algebraic semisimple group $G,$ for a class of discrete subgroups of $G$ that contains, for example, representations of a surface group on $\textrm{PSL}(2,\mathbb R)\times\textrm{PSL}(2,\mathbb R),$ induced by choosing two points on the Teichmüller space of the surface; and representations on the Hitchin component of $\textrm{PSL}(d,\mathbb R).$ We also prove a mixing property for the Weyl chamber flow in this setting.

math.GR↗

arXiv2012

Hyperconvex representations and exponential growth

Let $G$ be a real algebraic semi-simple Lie group and $Γ$ be the fundamental group of a compact negatively curved manifold. In this article we study the limit cone, introduced by Benoist, and the growth indicator function, introduced by Quint, for a class of representations $ρ:Γ\to G$ admitting a equivariant map from $\partialΓ$ to the Furstenberg boundary of $G$'s symmetric space together with a transversality condition. We then study how these objects vary with the representation.

math.GR↗