Simple groups separated by finiteness properties

TL;DR

This paper constructs simple groups with specific finiteness properties, demonstrating the existence of infinitely many quasi-isometry classes of finitely presented simple groups, expanding known examples beyond Kac--Moody groups.

Contribution

It introduces new simple groups with prescribed finiteness properties, derived from R"over--Nekrashevych groups, and shows they form infinitely many quasi-isometry classes.

Findings

Existence of simple groups of type F_{n-1} but not F_n for all n

First known examples for n ≥ 3 of such groups

Infinitely many quasi-isometry classes of finitely presented simple groups

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--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.