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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
  • 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 

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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