TY - JOUR
T1 - Anti-classification results for groups acting freely on the line
AU - Calderoni, Filippo
AU - Marker, David
AU - Motto Ros, Luca
AU - Shani, Assaf
N1 - Publisher Copyright: © 2023 Elsevier Inc.
PY - 2023/4/1
Y1 - 2023/4/1
N2 - We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings Ar(G) for a given countable group G, and prove that the equivalence relation induced by the natural action of GL2(Q) on Ar(Q2) is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau [29]. In particular, we show that its potential class is not Π30. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.
AB - We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings Ar(G) for a given countable group G, and prove that the equivalence relation induced by the natural action of GL2(Q) on Ar(Q2) is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau [29]. In particular, we show that its potential class is not Π30. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.
KW - Archimedean ordered groups
KW - Borel equivalence relations
KW - Borel reducibility
KW - Choiceless models
KW - Circularly ordered groups
KW - o-minimality
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U2 - 10.1016/j.aim.2023.108938
DO - 10.1016/j.aim.2023.108938
M3 - Article
SN - 0001-8708
VL - 418
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 108938
ER -