Elementary amenable groups and the space of marked groups

Abstract

The space of marked groups is a compact Polish space that
parameterizes all countable groups. This space allows for tools from
descriptive set theory to be applied to study group-theoretic
questions. The class of elementary amenable groups is the smallest
class that contains the abelian groups and the finite groups and that
is closed under group extension, taking subgroups, taking quotients,
and taking countable directed unions. In this talk, we first give a
characterization of elementary amenable marked groups in terms of
well-founded trees. We then show the set of elementary amenable marked
groups is coanalytic and non-Borel. This gives a new proof of a
theorem of Grigorchuk: There are finitely generated amenable
non-elementary amenable groups.