Finite Germ Extensions

2024/07/03 by James Belk, James Hyde, Belk, James +3
Mathematics · #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2407.03149

Abstract

We prove finiteness properties for groups of homeomorphisms that have finitely many "singular points", and we describe the normal structure of such groups. As an application, we prove that every countable abelian group can be embedded into a finitely presented simple group, verifying the Boone-Higman conjecture for countable abelian groups. Indeed, we describe a specific 2-generated, F_∞ simple group VA of homeomorphisms of the Cantor set that contains every countable abelian group. As a second application, we prove that if G is a bounded automata group then the associated Röver-Nekrashevych groups V_d,rG have type F_∞, verifying a conjecture of Nekrashevych for a large class of contracting self-similar groups. Among others, this result applies to Röver-Nekrashevych groups associated to Gupta-Sidki groups and the basilica group.

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• Embedding finitely presented self-similar groups into finitely presented simple groups
• Infinitely presented simple groups separated by homological finiteness properties
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