DynGeo

The rôle of infinite discrete subgroups of Lie groups originated in Fuchsian equations and in crystallographic groups, and gradually grew as its arithmetical, ergodical, dynamical, and geometrical aspects developed along the years. The objective of the 4 years project "Dynamics and geometric structures" is to uncover new and remarkable relations between the dynamical and geometrical facets of those groups. This includes a systematic investigation of the generalization of Fuchsian groups, the study of the advanced structure of their moduli space coming from the thermodynamic formalism as well as the intertwinings of geometrical and analytical properties of space-times with their infinity. The project has been organized along five different but interrelated directions we now present. Anosov representations: it is has long been an unresolved question to find the class of discrete groups that should be the higher rank generalization to Fuchsian or convex-cocompact subgroups. It is now widely agreed that the class of Anosov representations invented by Labourie answers this question. Anosov representations have now been characterized in a number of ways. The team wants to address the question of finding the higher rank equivalent of geometrically finite subgroups and also to investigate properties of dynamical systems associated with discrete subgroups. Homogeneous geometry: since the early developments of hyperbolic geometry, the connections between geometry and discrete groups are many. Recent illustrations are the way how Anosov subgroups give rise to geometric structure and the understanding of Lorentzian manifolds of constant curvature. One aim of the project is to explore furthermore those connections. More specifically, the foreseen tasks are to parametrize some moduli spaces of representations using geometric coordinates, to understand compactifications of Riemannian locally symmetric spaces but also to explore Lorentzian manifolds "with particles" which is physically more relevant. Length spectrum: any discrete subgroup gives rise to a length spectrum that in general determines completely the discrete subgroup. In the spirit of Thurston's asymmetric metric on Teichmüller spaces, the team is going to examine the length spectrum comparison in the setting of Anosov representations, and particularly for the Hitchin components, a generalization of the Teichmüller component due to Hitchin. The entropy is the exponential growth rate of the length spectrum, the teams is going to bring to light rigidity results of the entropy as well as its local and global behaviors in that context of "higher Teichmüller spaces". Pressure metric: the above mentioned length spectrum is often realized as the closed orbits lengths of a flow and hence topological and dynamical invariants of this flow can be accessed. Among those is its pressure and thus there is an associated pressure metric on the deformation space of representations. The local behavior of that metric as well as the other numerical quantities (lengths, intersections numbers, etc.) are among the scheduled subjects. Renormalized volume: the renormalized volume is a way to define a "volume" in a context where the volume is infinite. Is has been well studied in the context of convex cocompact hyperbolic 3-manifolds and has strong links with the geometry of the Teichmüller space; in the context of quasifuchsian groups it is related to the Liouville action. The related questions the team wants to inquiry in depth are the fine geometry of hyperbolic 3-manifolds, the possibility to define the Liouville action for the Hitchin component and also to determine a renormalization procedure for the Hitchin component themselves. The project is organized around 5 partners: Lille, Luxembourg, Nice, Paris, and Strasbourg.