Simple groups separated by finiteness properties

2017/12/14 by Rachel Skipper, Skipper, Rachel, Stefan Witzel +3 · 7 citations
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1712.05361

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

arxiv created 2018/10/22 · arxiv updated 2018/10/23

Abstract

We show that for every positive integer n there exists a simple group that is of type F_n-1 but not of type F_n. For n≥ 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.

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