  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.
