Dynamics and geometric structures – 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.