Groups, Geometry, and Dynamics

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Volume 11, Issue 2, 2017, pp. 649–684
DOI: 10.4171/GGD/411

Published online: 2017-06-26

Chain conditions, elementary amenable groups, and descriptive set theory

Phillip Wesolek[1] and Jay Williams[2]

(1) Université Catholique de Louvain, Louvain-La-Neuve, Belgium
(2) California Institute of Technology, Pasadena, USA

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 descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees, and therefore, elementary amenability is equivalent to a chain condition. Our characterization again implies the set of elementary amenable groups is co-analytic and non-Borel. As corollary, we obtain a new, non-constructive, proof of the existence of finitely generated amenable groups that are not elementary amenable.

Keywords: Chain conditions, elementary amenable groups, descriptive set theory, space of marked groups

Wesolek Phillip, Williams Jay: Chain conditions, elementary amenable groups, and descriptive set theory. Groups Geom. Dyn. 11 (2017), 649-684. doi: 10.4171/GGD/411