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Localization, metabelian groups, and the isomorphism problem


Authors: Gilbert Baumslag, Roman Mikhailov and Kent E. Orr
Journal: Trans. Amer. Math. Soc. 369 (2017), 6823-6852
MSC (2010): Primary 20F14, 20F16; Secondary 20F05, 20F10
DOI: https://doi.org/10.1090/tran/6838
Published electronically: March 1, 2017
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Abstract: If $ G$ and $ H$ are finitely generated, residually nilpotent metabelian groups, $ H$ is termed para-$ G$ if there is a homomorphism of $ G$ into $ H$ which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties, including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and others where this is not the case. We give several examples where we solve the Isomorphism Problem. We prove also that the sequence of torsion-free ranks of the lower central quotients of a finitely generated metabelian group is computable. In a future paper we plan on proving that there is an algorithm to compute the numerator and denominator of the rational Poincaré series of a finitely generated metabelian group and will carry out this computation in a number of examples, which may shed a tiny bit of light on the Isomorphism Problem. Our proofs use localization, class field theory and some constructive commutative algebra.


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Additional Information

Gilbert Baumslag
Affiliation: CAISS and Department of Computer Science, City College of New York, Convent Avenue and 138th Street, New York, New York 10031

Roman Mikhailov
Affiliation: Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia – and – St. Petersburg Department of the Steklov Mathematical Institute, Fontanka 27, Saint Petersburg, 191023 Russia
Email: rmikhailov@mail.ru

Kent E. Orr
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: korr@indiana.edu

DOI: https://doi.org/10.1090/tran/6838
Received by editor(s): September 29, 2014
Received by editor(s) in revised form: September 27, 2015
Published electronically: March 1, 2017
Additional Notes: The research of the first author was supported by Grant CNS 111765, and the work done here was initially carried out at IHES, whose hospitality is gratefully acknowledged
The research of the second author was supported by Saint-Petersburg State University research grant N 6.37.208.2016 and by JSC “Gazprom Neft”.
The third author thanks the Simons Foundation, Grants 209082 and 4429401, for their support.
Article copyright: © Copyright 2017 American Mathematical Society