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Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions
Abstract
We prove that if a group G contains a Baumslag-Solitar group BS(m,n), with mn≠0, then the theory of fields with a G-action has no model companion. In particular, every group containing ℤ2≅BS(1,1) has no model companion for its field actions, resolving a conjecture of Beyarslan and Kowalski. Consequently, there are no model companions for field actions of ℚ2, Thompson's groups F,T,V, or Higman's group. Our argument adapts Hrushovski's method for two commuting automorphisms. The key modification is the cyclotomic condition θq, which removes the need to prescribe a non-trivial action on a root of unity and thereby makes Hrushovski's method applicable after passing from a subgroup to an ambient group.
Related
- On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II
- On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and SLn(Z)
- Singular Loci of SL2(C)-Representation Varieties of Non-Hopfian Baumslag--Solitar Groups
- The measure equivalence classification of Baumslag-Solitar groups
- Surface subgroups of Baumslag doubles along short words
BibTeX
@misc{yanagawa2026baumslagsolitar,
title = {{Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions}},
author = {Makoto Yanagawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01996},
eprint = {2609.01996},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01996},
}
RIS
TY - UNPB
TI - Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions
AU - Makoto Yanagawa
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.01996
UR - https://arxiv.org/abs/2609.01996
AB - We prove that if a group G contains a Baumslag-Solitar group BS(m,n), with mn≠0, then the theory of fields with a G-action has no model companion. In particular, every group containing ℤ^2≅BS(1,1) has no model companion for its field actions, resolving a conjecture of Beyarslan and Kowalski. Consequently, there are no model companions for field actions of ℚ^2, Thompson's groups F,T,V, or Higman's group. Our argument adapts Hrushovski's method for two commuting automorphisms. The key modification is the cyclotomic condition θ_q, which removes the need to prescribe a non-trivial action on a root of unity and thereby makes Hrushovski's method applicable after passing from a subgroup to an ambient group.
ER -