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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The coarse classification of countable abelian groups


Authors: T. Banakh, J. Higes and I. Zarichnyi
Journal: Trans. Amer. Math. Soc. 362 (2010), 4755-4780
MSC (2010): Primary 20F65; Secondary 57M07, 20F69
Published electronically: April 27, 2010
MathSciNet review: 2645049
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Abstract: We prove that two countable locally finite-by-abelian groups $ G,H$ endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely generated or both are infinitely generated. On the other hand, we show that each countable group $ G$ that coarsely embeds into a countable abelian group is locally nilpotent-by-finite. Moreover, the group $ G$ is locally abelian-by-finite if and only if $ G$ is undistorted in the sense that $ G$ can be written as the union $ G=\bigcup_{n\in\omega}G_n$ of countably many finitely generated subgroups such that each $ G_n$ is undistorted in $ G_{n+1}$ (which means that the identity inclusion $ G_n\to G_{n+1}$ is a quasi-isometric embedding with respect to word metrics on $ G_n$ and $ G_{n+1}$).


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Additional Information

T. Banakh
Affiliation: Instytut Matematyki, Akademia Świȩtokrzyska w Kielcach, Poland – and – Department of Mathematics, Ivan Franko National University of Lviv, Ukraine
Email: tbanakh@yahoo.com

J. Higes
Affiliation: Departamento de Geometría y Topología, Facultad de CC.Matemáticas, Universidad Complutense de Madrid, Madrid, Spain
Address at time of publication: Institute Mathematics, MA 6-2, Technische Universität Berlin, 10623, Berlin, Germany
Email: josemhiges@yahoo.es

I. Zarichnyi
Affiliation: Department of Mathematics, Ivan Franko National University of Lviv, Ukraine
Email: ihor.zarichnyj@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05118-4
PII: S 0002-9947(10)05118-4
Keywords: Coarse geometry, countable abelian groups, asymptotic dimension
Received by editor(s): October 21, 2008
Published electronically: April 27, 2010
Additional Notes: The second named author was supported by Grant AP2004-2494 from the Ministerio de Educación y Ciencia, Spain and project MEC, MTM2006-0825. He thanks Kolya Brodskyi and A. Mitra for helpful discussions. He also thanks Jose Manuel Rodriguez Sanjurjo for his support, and gives special thanks to Jerzy Dydak for all his help and very nice suggestions.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.