Chain conditions, elementary amenable groups, and descriptive set theory

5 F eb 2 01 5 CHAIN CONDITIONS , ELEMENTARY AMENABLE GROUPS , AND DESCRIPTIVE SET THEORY

We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptiveset-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions...

Descriptive Set Theory Descriptive Set Theory

Non-elementary amenable subgroups of automata groups

We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups ...

Recurrent Points and Discrete Points for Elementary Amenable Groups

Let fig be the Stone-Cech compactification of a group G, Aa the set of all almost periodic points in G, Ka c[U { supp eLIM(G)}] and Ra the set of all recurrent points in fiG. In this paper we will study the relationships between Ka and Ra, and between Aa and Ra. We will show that for any infinite elementary amenable group G, Aa Ra and RaKa =/= .

Descriptive Set Theory Problem Set

Prove that any strictly monotone sequence (Uα)α<γ of open subsets of X has countable length, i.e. γ is countable. Hint: Use the same idea as in the proof of (a). (c) Show that every monotone sequence (Uα)α<ω1 open subsets of X eventually stabilizes, i.e. there is γ < ω1 such that for all α < ω1 with α ≥ γ, we have Uα = Uγ. Hint: Use the regularity of ω1. (d) Conclude that parts (a), (b) and (c)...