Simple groups separated by finiteness properties

@article{Skipper2018SimpleGS,
  title={Simple groups separated by finiteness properties},
  author={Rachel Skipper and Stefan Witzel and Matthew C. B. Zaremsky},
  journal={Inventiones mathematicae},
  year={2018},
  volume={215},
  pages={713-740}
}
We show that for every positive integer n there exists a simple group that is of type $$\mathrm {F}_{n-1}$$Fn-1 but not of type $$\mathrm {F}_n$$Fn. For $$n\ge 3$$n≥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… 

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