The strong Atiyah and Lück approximation conjectures for one-relator groups

  • Andrei Jaikin-ZapirainEmail author
  • Diego López-Álvarez


It is shown that the strong Atiyah conjecture and the Lück approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.



This paper is partially supported by the Grant MTM2017-82690-P of the Spanish MINECO and by the ICMAT Severo Ochoa project SEV-2015-0554. We would like to thank Fabian Henneke and Dawid Kielak for suggesting that our argument can be used to prove that the class of torsion-free groups satisfying the Atiyah conjecture is closed under extensions by locally indicable groups. We are also very grateful to Javier Sánchez for useful discussions, to Fabian Henneke for many comments on a previous version of this paper and to an anonymous referee for a thorough reading of the paper and useful suggestions.


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Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  2. 2.Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCMMadridSpain

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