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Published 2017 | Submitted
Journal Article Open

Chain conditions, elementary amenable groups, and descriptive set theory

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.

Additional Information

© 2017 European Mathematical Society. Received February 12, 2015. We would like to thank Alexander Kechris and Andrew Marks for helpful mathematical discussions. We also thank François Le Maître for his insightful remarks. J. Williams was partially supported by NSF Grant 1044448, Collaborative Research: EMSW21-RTG: Logic in Southern California.

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August 19, 2023
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October 17, 2023