Simple groups separated by finiteness properties

By Rachel Skipper, Stefan Witzel, Matthew C. b. Zaremsky. Inventiones Mathematicae, 2017

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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 groups, and contain free abelian groups of infinite rank.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Geometry and Topology

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