Finite Germ Extensions

TL;DR

This paper establishes finiteness properties and structural results for groups of homeomorphisms with finitely many singular points, embedding countable abelian groups into finitely presented simple groups and analyzing automata groups.

Contribution

It proves the Boone-Higman conjecture for countable abelian groups and shows certain automata groups have type F_infinity, introducing new simple groups of homeomorphisms.

Findings

Every countable abelian group embeds into a finitely presented simple group.

Certain automata groups have type F_infinity.

Constructs specific simple groups of homeomorphisms of the Cantor set.

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 R\"over-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 R\"{o}ver-Nekrashevych groups associated to Gupta-Sidki groups and the basilica group.