The coarse classification of countable abelian groups
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- by T. Banakh, J. Higes and I. Zarichnyi PDF
- Trans. Amer. Math. Soc. 362 (2010), 4755-4780 Request permission
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 $G_{n+1}$).References
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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
- MR Author ID: 249694
- 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
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4755-4780
- MSC (2010): Primary 20F65; Secondary 57M07, 20F69
- DOI: https://doi.org/10.1090/S0002-9947-10-05118-4
- MathSciNet review: 2645049