We introduce a family of groups of homeomorphisms obtained from groups of piecewise linear homeomorphisms by adding finitely many singularities and we prove results about their simplicity, abelianizations and finiteness properties. This family arose naturally in the process of solving a Kourovka notebook question by Bridson and De la Harpe asking whether there exists a finitely presented group containing the additive group Q of rational numbers.
Among the examples we construct, we describe two groups TA and VA that are simple, two-generated, finitely presented and contain, respectively, all countable torsion-free abelian groups and all countable abelian groups, explicitly realizing the Boone-Higman embedding theorem. Moreover, we show that they have type F_\infty (a generalization of finite presentability).
We also discuss how our results can be applied to other related groups, such as some Nekrashevych groups, a class of groups which are generated by Thompson groups V_{n,r} and suitable self-similar groups
