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--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.

arXiv comment25 pages. v2: incorporated comments v3: final version, to appear, Invent. Math