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Abstract:
The subject of amenability essentially begins in 1900's with
Lebesgue. He asked whether the properties of his integral are really
fundamental and follow from more familiar integral axioms. This led
to the study of positive, finitely additive and translation invariant
measure on reals as well as on other spaces. In particular the study
of isometry-invariant measure led to the Banach-Tarski decomposition
theorem in 1924. The class of amenable groups was introduced by von
Neumann in 1929, who explained why the paradox appeared only in
dimensions greater or equal to three, and does not happen when we
would like to decompose the two-dimensional ball. In 1940's, M. Day
formally defined a class of elementary amenable groups as the largest
class of groups amenability of which was known to von Naumann. He
asked whether there are other groups then that. Currently there are
many groups that answer von Neumann-Day's question. However, in each
particular case it is algebraically difficult to show that the group
is not elementary amenable, and analytically difficult to show that it
is amenable. The talk is aimed to discuss recent developments and
approaches in the field. In particular, it will be shown how to prove
amenability of all known non-elementary amenable groups using only one
single approach. We will also discuss techniques coming from random
walks of groups.
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