In his seminal book “Trees” Serre laid down the fundamentals of the theory of groups acting on simplicial trees. In the following decade Serre's novel approach unified several geometric, algebraic, and combinatorial methods of group theory into a unique powerful tool, known today as Bass-Serre Theory. Topologists became interested in R-trees with the work of Morgan and Shalen (1985) which generalized parts of Thurston's Geometrization Theorem. A joint effort of several researchers culminated in a description of finitely generated groups acting freely on R-trees, which is now known as Rips' theorem. The key ingredient of the theory is the so-called “Rips machine”. The idea of the Rips machine comes from Makanin's algorithm (or elimination process) for solving equations in free groups.
Actions on R-trees effectively cover all Archimedean actions, since every group acting freely on a λ-tree for an Archimedean ordered abelian group λ acts freely also on an R-tree. The case of non-Archimedean free actions is wide open. Bass studied finitely generated groups acting freely on a λ ⊕ Z-tree with respect to the lexicographic order on λ ⊕ Z. Recently, Guirardel studied finitely generated groups acting freely on an R-tree (with the lexicographic order). However, the main problem of the Alperin-Bass-Rips program, which asks for a description of finitely generated groups acting freely on λ-trees, remains largely open.
One of the main goals of this workshop is to focus on the various approaches to this problem. In particular, we are going to study relations between the Makanin-Razborov elimination process and symbolic dynamics, especially methods of the interval exchange theory.Another goal will be to develop newly discovered connections between the theory of groups acting on rooted trees and dynamical systems. On the one hand, conformal dynamical systems yield a rich source of such groups with interesting algebraic, geometric, and spectral properties. On the other hand, such groups can be shown to yield conformal dynamical systems with remarkable geometric and measure-theoretic regularity. This connection is established by means of the theory of Gromov hyperbolic spaces.
Groups acting on R-trees
M. Bestvina (Utah)
V. Nekrashevich (Texas A&M)
Generalizations of relative hyperbolicity
D. Osin (CUNY)