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arXiv · 2407.03149

Finite Germ Extensions

James Belk·James Hyde·Francesco Matucci
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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, $\mathrm{F}_\infty$ simple group $V\mathcal{A}$ 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 $V_{d,r}G$ have type $\mathrm{F}_\infty$, 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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James Belk, James Hyde, Francesco Matucci. 2024-07-03. Finite Germ Extensions. https://arxiv.org/abs/2407.03149

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