Les Journées GAAG (Géométrie en Action et Actions en Géométrie) rassemblent des chercheuses et chercheurs dont les travaux impliquent crucialement des actions de groupes discrets sur certains espaces géométriques. Elles sont organisées conjointement par le Département de Mathématiques de l’Université du Luxembourg, l’IRMA Strasbourg, l’IHES et l’IECL Metz-Nancy.
Correspondants actuels : Charles Frances (Strasbourg), Fanny Kassel (IHES), Jean-Marc Schlenker (Luxembourg), Samuel Tapie (Nancy).
Journée GAAG du 28 janvier 2026 : IECL Nancy
Programme à venir
Journée GAAG du 16 septembre 2025 : IECL Nancy
11h accueil salle Döblin (4ème étage)
11h30 – 12h15 : Pierre-Louis Blayac, L’espace de Teichmüller est distordu dans l’espace de Hitchin, salle Döblin
14h – 14h45 : Samuel Tapie (IECL Nancy), Mixing of the geodesic flow for actions with a contracting element, salle de conférences (2ème étage)
15h – 15h45 : Alex Nolte (IHES), Hyperconvexity, canonical parameterizations, and the geometry of Hitchin representations, salle de conférences (2ème étage)
16h15 – 17h : Wayne Lam (Luxembourg), Deformation Spaces of Circle Patterns, salle de conférences (2ème étage)
Résumés :
Pierre-Louis Blayac :
L’espace de Teichmüller est distordu dans l’espace de HitchinÉtant donnée une surface S de genre au moins 2 et un entier d>2, l’espace des métriques hyperboliques marquées, dit espace de Teichmüller T(S), peut se voir comme l’espace des représentations du groupe fondamentale de S dans PSL(2,R) modulo conjugaison. En postcomposant par la représentation irréductible de PSL(2,R) dans PSL(d,R), T(S) se plonge dans les représentations à valeurs dans PSL(d,R). La composante connexe contenant T(S) est la composante de Hitchin H(S), qui est homéomorphe à un espace vectoriel et constituée de représentations fidèles et discrètes par les travaux de Hitchin, Labourie, Fock et Goncharov.
Wayne Lam : Deformation Spaces of Circle Patterns
William Thurston proposed viewing the map induced by two circle packings in the plane with the same tangency pattern as a discrete conformal map.
A discrete analogue of the Riemann mapping theorem then follows from the Koebe–Andreev–Thurston theorem. A natural question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete ones. Since circles are preserved under complex projective transformations, it is natural to consider circle packings on surfaces endowed with complex projective structures. Kojima, Mizushima, and Tan conjectured that, for a given combinatorics, the deformation space of
circle packings is homeomorphic to Teichmüller space.
In this talk, we present recent progress on this conjecture and describe its various connections.
Alex Nolte : Hyperconvexity, canonical parameterizations, and the geometry of Hitchin representations
PSL(n,R)-Hitchin representations are representations of surface groups in PSL(n,R) with strong convexity properties, and play a major role in higher-rank Teichmüller theory. An interesting property of Hitchin epresentations is that they induce natural parameterizations of many distinguished subsets of flag manifolds. I will present a pair of results related to this phenomenon. First, I will explain how such maps give rise to geometric models for previously studied reparameterizations of the geodesic flow on a hyperbolic surface. Then, I will explain how these maps can be used to understand the shape of domains of discontinuity for Hitchin representations in PSL(4,R) and state a rigidity result on group invariant foliations that is proved with this structure. The first part is joint work with Max Riestenberg.
Samuel Tapie : Mixing of the geodesic flow for action with a contracting element
Let (X,d) be a proper geodesic metric space and G be a discrete group acting by isometries. Informations on the action are given by its growth function, i.e. the map
$N_G(x,R) := \#\{ g\in G \; ; \; d(x, g.x) \leq R\}.$
To obtain precise asymptotic estimates on this growth function, many strategies have been developped when (X,d) is the real hyperbolic space $\mathbb H^n$. Among these methods, a very robust (adapted in non-smooth CAT(-1) settings or in some CAT(0) settings) relies on the mixing of the geodesic flow for suitable invariantmeasures (Bowen-Margulis-Sullivan measures).
In this talk, we will describe a very general construction of a « geodesic flow » for discrete actions, which we call « horo-geodesic flow », how it relates to previously known geodesic flows and why it is mixing for its Bowen-Margulis-Sullivan measures as soon as the action has a contracting (or rank one) element. This is joint work with Rémi Coulon.
Pour toute information pratique, merci de contacter Samuel Tapie.