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Anne Parreau

Publications and source records attributed to Anne Parreau.

10 recordsLinked to original sources
arXiv2025

Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries

We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric.

math.MG↗
arXiv2023

The real spectrum compactification of character varieties

We study the real spectrum compactification of character varieties of finitely generated groups in semisimple Lie groups. This provides a compactification with good topological properties, and we interpret the boundary points in terms of actions on building-like spaces. Among the applications we give a general framework guaranteeing the existence of equivariant harmonic maps in building-like spaces.

math.GR↗
arXiv2021

Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $Γ$ of a closed hyperbolic surface $Σ$ in ${\rm PSL}(2,\mathbb R)^n$. We identify the boundary with the sphere $\mathbb P((\mathcal{ML})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $Σ$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of $\mathbb R^2$-mixed structures on $Σ$. We associate to such a structure a dual tree-graded space endowed with an $\mathbb R_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb R$-trees with the given length spectrum.

math.GT↗
arXiv2021

Positive crossratios, barycenters, trees and applications to maximal representations

We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat-Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters.

math.GT↗
arXiv2019

On triples of ideal chambers in A2-buildings

We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of generic triples of ideal chambers of X and relate it with the triple ratio of triples of flags.

math.MG↗
arXiv2017

A structure theorem for geodesic currents and length spectrum compactifications

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bilipschitz equivalent to the length function with respect to a hyperbolic metric. We show that the subset of currents with positive systole is open and that the mapping class group acts properly discontinuously on it. As an application, we obtain in the case of compact surfaces a structure theorem on the length functions appearing in the length spectrum compactification both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group.

math.GT↗
arXiv2015

Invariant subspaces for some surface groups acting on A2-Euclidean buildings

This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Archimedean representations acting on the Euclidean building preserve a cocompact weakly convex subspace, which is part flat surface and part tree. In particular the eigenvalue and length(s) spectra are given by an explicit finite A2-complex. We use this result to describe degenerations of real convex projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parametrization in A2-buildings.

math.GT↗
arXiv2011

Compactification d'espaces de représentations de groupes de type fini

Let $Γ$ be a finitely generated group and $G$ be a noncompact semisimple connected real Lie group with finite center. We consider the space $\mathcal X$ of conjugacy classes of reductive representations of $Γ$ into $G$. We define the {\it translation vector} of an element $g$ in $G$, with values in a Weyl chamber, as a refinement of the translation length in the associated symmetric space. We construct a compactification of $\mathcal X$, induced by the marked translation vector spectrum, generalizing Thurston's compactification of the Teichmüller space. We show that the boundary points are projectivized marked translation vector spectra of actions of $Γ$ on affine buildings with no global fixed point. An analoguous result holds for any reductive group $G$ over a local field.

math.DG↗
arXiv2010

Espaces de représentations complètement réductibles

We study some geometric properties of actions on nonpositively curved spaces related to complete reducibility and semisimplicity, focusing on representations of a finitely generated group in the group G of rational points of a reductive group over a local field, acting on the associated space (symmetric space or affine building). We prove that the space of completely reducible classes is the maximal Hausdorff quotient space for the conjugacy action of G on the representation space.

math.GR↗