Growth in groups and amenability
Abstract: The volume of a geodesic ball of radius tending to infinity grows polynomially in case of Euclidian space and grows exponentially in case of hyperbolic space. It is difficult to construct (and even to imagine) a Riemannian manifold with a large group of isometries which would have an intermediate growth between polynomial and exponential. But such manifolds exist and can be constructed by help of groups of intermediate growth. The growth in groups is a subject initiated by A. Schwartz, J. Milnor, J. Wolf and H. Bass. It plays an important role in asymptotic group theory and its applications. The groups of intermediate growth constructed by the speaker in 1983 (as the answer to a question of Milnor) opened a new direction in the study of groups acting on trees including groups generated by finite automata. Also a class of branch group was defined and investigated. In the first half of the talk we shall outline the current situation in the study of growth in groups.
The notion of amenable group was introduced by von Neumann with the purpose to understand the nature of the Hausdorff-Banach-Tarsky paradox, one of the form of which states that the unit ball in three-dimensional Euclidian space can be split into finitely many pieces which can be rearranged using isometries into a new figure that is a disjoint union of two balls of unit radius. The alternative: amenable - nonamenable is very important in many considerations and the notion enters such areas of mathematics as dynamical systems, statistical mechanics, random walks, operator algebras, Riemannian geometry, theory of representations etc. For instance, still open is a long standing Conjecture of Dixmier claiming that amenability of a group is equivalent to the property that any uniformly bounded representation is similar to a unitary representation. There were two main attempts to describe the class of amenable groups: firstly as the class EG of elementary amenable groups i.e. groups that can be constructed from finite and commutative groups by operations of extensions and direct limits, and secondly as the class NF of groups without free subgroup with two generators. Both attempts failed as it was discovered that the class of amenable groups is much larger than the class EG but is smaller than the class NF. We shall describe some previous results concerning this area and propose new examples that are potential candidates to answer several open questions.