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Chain conditions, elementary amenable groups, and descriptive set theory

Wesolek, Phillip and Williams, Jay (2017) Chain conditions, elementary amenable groups, and descriptive set theory. Groups, Geometry, and Dynamics, 11 (2). pp. 649-684. ISSN 1661-7207. doi:10.4171/GGD/411.

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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.

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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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Subject Keywords:Chain conditions, elementary amenable groups, descriptive set theory, space of marked groups
Issue or Number:2
Classification Code:Mathematics Subject Classification (2010): Primary: 20E34, Secondary: 54H05
Record Number:CaltechAUTHORS:20170831-123607939
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Official Citation: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
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81017
Deposited By: Tony Diaz
Deposited On:31 Aug 2017 20:31
Last Modified:15 Nov 2021 19:40

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