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Inventiones mathematicae

, Volume 215, Issue 2, pp 713–740 | Cite as

Simple groups separated by finiteness properties

  • Rachel SkipperEmail author
  • Stefan Witzel
  • Matthew C. B. Zaremsky
Article
First Online:
  • 154 Downloads

Abstract

We show that for every positive integer n there exists a simple group that is of type \(\mathrm {F}_{n-1}\) but not of type \(\mathrm {F}_n\). For \(n\ge 3\) these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace–Rémy, consists of non-affine Kac–Moody groups over finite fields. Our examples arise from Röver–Nekrashevych groups, and contain free abelian groups of infinite rank.

Mathematics Subject Classification

Primary 20E32 Secondary 57M07 20F65 20E08 
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Notes

Acknowledgements

We are grateful to Dessislava Kochloukova and Said Sidki for sharing a preprint of [36] with us and to Bertrand Rémy for asking the question that motivated our search for simple groups separated by finiteness properties. We also thank Matt Brin, Pierre-Emmanuel Caprace, Eduard Schesler, and Marco Varisco for many helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Rachel Skipper
    • 1
    Email author
  • Stefan Witzel
    • 2
  • Matthew C. B. Zaremsky
    • 3
  1. 1.Mathematics InstituteUniversity of GöttingenGöttingenGermany
  2. 2.Department of MathematicsBielefeld UniversityBielefeldGermany
  3. 3.Department of Mathematics and StatisticsUniversity at Albany (SUNY)AlbanyUSA

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