JournalsggdVol. 11, No. 2pp. 649–684

Chain conditions, elementary amenable groups, and descriptive set theory

  • Phillip Wesolek

    Université Catholique de Louvain, Louvain-La-Neuve, Belgium
  • Jay Williams

    California Institute of Technology, Pasadena, USA
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Abstract

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.

Cite this article

Phillip Wesolek, Jay Williams, Chain conditions, elementary amenable groups, and descriptive set theory. Groups Geom. Dyn. 11 (2017), no. 2, pp. 649–684

DOI 10.4171/GGD/411