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.