Simple groups separated by finiteness properties

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