Simple groups separated by finiteness properties

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–Remy, consists of non-affine Kac–Moody groups over finite fields. Our examples arise from Rover–Nekrashevych groups, and contain free abelian groups of infinite rank.