Math Seminars
Rostislav Grigorchuk
Texas A&M University
Progress in the Study of the Alternative "Amenable/Nonamenable"
During the Past 50 Years.
The notion of an amenable group was introduced (under the name
measurable) by von Neumann in 1929 with a purpose of understandings
the roots of the phenomenon known as Banach-Tarskii Paradox. The notion
was studied and used in different branches of mathematics by many
prominent mathematicians (Tarskii, Bogolyubov). Staring from the
1950's it began to play an important role in the theory of dynamical
systems and different versions of this notion are the subject of current
research in Ergodic Theory. There are hundreds of equivalent definitions
of the class of amenable groups , but we are still very far from having
a complete picture which describes which groups are amenable and which
are not.
In my talk I will focus on a series of results related to two Problems
of M.Day from 1957, one of which is known as von Neumann Conjecture. I
will explain how these problems were solved and also present the history
of the solutions to some other problems in the same spirit. Some of the
results are very recent.
The classes of groups called Branch Groups and Finite Automata
Groups will play an essential role in the exposition. These classes are
important also from dynamical point of view because interesting
dynamical systems can be associated to them (the simplest one is the
well known Adding Machine).