The methods which have originated from dynamical systems on one hand and from geometry on the other have become central in present day group theory.
The theorem of Rips-Bestvina-Paulin constitutes an emblematic occurrence where geometric and dynamical methods are blended together to prove a deep result in group theory: the group of outer automorphisms of a hyperbolic group G without torsion is infinite if and only if G splits over an infinite cyclic group. Indeed, the idea of the proof is to use first the hyperbolicity of G in order to derive, via a renormalization process, an action of G on an R-tree T. In a second step this action has to be analyzed sufficiently precisely: the "Rips machine" allows to approximate the action of G on T with actions of G an simplicial trees, both at a metric and a dynamical level.
More generally, R-trees have played a key role in geometric group theory during the last 25 years: in particular, they appear as natural limit objects in the degeneration of hyperbolic structures (used for instance to compactify spaces such as the Teichmüller spaces).
By now, the areas of dynamics and of geometric group theory extend to numerous active branches of mathematics such as low-dimensional topology, algebraic topology, complex dynamics, Teichmüller theory, logics, Riemannian geometry, representation theory, operator algebras... The present conference will give the opportunity not only to present the most recent progress in dynamics of group and/or geometric group theory, but also to report on their recent impact on related domains.
Besides, during this conference we will celebrate the sixtieth birthday of two mathematicians
– Gilbert Levitt, Université de Caen,
– Martin Lustig, Université Aix-Marseille,
whose work on the dynamics and geometry of groups have contributed importantly to the development of this area in the past 30 years (in particular through their joined work on R-trees and automorphisms of free groups).
