Simple groups separated by finiteness properties

Skipper R, Witzel S, Zaremsky MCB. Simple groups separated by finiteness properties. INVENTIONES MATHEMATICAE. 2019;215(2):713-740.We show that for every positive integer n there exists a simple group that is of type Fn-1 but not of type Fn. For n3 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