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.
Classification :
20-XX, 54-XX
Mots-clés :
Chain conditions, elementary amenable groups, descriptive set theory, space of marked groups
Affiliations des auteurs :
Phillip Wesolek
1
;
Jay Williams
2
1
Université Catholique de Louvain, Louvain-La-Neuve, Belgium
2
California Institute of Technology, Pasadena, USA
Phillip Wesolek; Jay Williams. Chain conditions, elementary amenable groups, and descriptive set theory. Groups, geometry, and dynamics, Tome 11 (2017) no. 2, pp. 649-684. doi: 10.4171/ggd/411
@article{10_4171_ggd_411,
author = {Phillip Wesolek and Jay Williams},
title = {Chain conditions, elementary amenable groups, and descriptive set theory},
journal = {Groups, geometry, and dynamics},
pages = {649--684},
year = {2017},
volume = {11},
number = {2},
doi = {10.4171/ggd/411},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/411/}
}
TY - JOUR
AU - Phillip Wesolek
AU - Jay Williams
TI - Chain conditions, elementary amenable groups, and descriptive set theory
JO - Groups, geometry, and dynamics
PY - 2017
SP - 649
EP - 684
VL - 11
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/411/
DO - 10.4171/ggd/411
ID - 10_4171_ggd_411
ER -
%0 Journal Article
%A Phillip Wesolek
%A Jay Williams
%T Chain conditions, elementary amenable groups, and descriptive set theory
%J Groups, geometry, and dynamics
%D 2017
%P 649-684
%V 11
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/411/
%R 10.4171/ggd/411
%F 10_4171_ggd_411
