Simple groups separated by finiteness properties

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

paper · doi:10.48550/arxiv.1712.05361

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.

Cited by

• Finitely presented simple groups with no piecewise projective actions
• Boundary actions of outer automorphism groups of Thompson-like groups
• Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups
• Embedding groups into acyclic groups
• Infinitely presented simple groups separated by homological finiteness properties
• Chern character for torsion-free ample groupoids
• Simple totally disconnected locally compact groups separated by finiteness properties

Related

• Infinitely presented simple groups separated by homological finiteness properties
• Finitely presented simple groups with no piecewise projective actions
• An incoherent simple group
• Infinite simple characteristic quotients
• Quasi-simple finite groups of essential dimension 3