Facets of Discrete Groups. – DiscGroup

Facets of discrete groups

Geometric group theory involves many viewpoints, including geometric, dynamical, probabilistic, analytic ones. The goal of the project is to gather researchers with diverse approaches on the subject and to allow them to share ideas among them.

Sutdy of discrete groups from the point of view of model theory, growth, of their representations, of negative curvature aspects and of fundamental groups of manifolds

We propose to investigate the deep connections revealed by Sela that exist between the first-order logic of a group and its geometry by considering other notions of model theory (definable sets, forking, independence...),<br />and by enlarging the classes of groups studied (relatively hyperbolic groups, groups with nonpositive curvature). We also propose to study the structure of equations and the isomorphism problem, and to extend the small cancellation theory in such classes groups.<br /><br />We propose to understand whether the relations between the volume of a hyperbolic 3 manifold and the combinatorial complexity of its fundamental group still exist in higher dimensions.<br />We would like to apply geometric group theory techniques to study Kähler groups. For instance, we propose to study fundamental groups of manifolds obtained as ramified covers, and understand when they are hyperbolic.<br /><br />We want to study a strictly convex projective manifold from a dynamical point of view. The goal is to show that the quotient of a strictly convex open set is of finite volume if and only if every point in the boundary is a bounded parabolic fixed point or a conical limit point.<br /><br />Another project is to investigate the geometry of compact Anti-deSitter manifolds and to give a combinatorial description in terms of geodesic graphs maximally stretched by Lipschitz maps.<br /><br />We propose to study which functions can be realized as a growth function of a group. We mention that a conjecture due to Grigorchuk says that there is a gap in the possible growth function between polynomials and exponential of a square root of n. We would also like to understand approximate groups in groups of geometric origins having a slow growth behaviour. In the specific group of interval exchange transformations, we also want to investigate the possible existence of groups of intermediate growth.