Geometric orbifolds with torsion free derived subgroup

A geometric orbifold of dimension $d$ is the quotient space $\mathscr O=X/K$, where $(X,G)$ is a geometry of dimension $d$ and $K<G$ is a co-compact discrete subgroup. In this case $\pi^\mathrm{orb}_1(\mathscr O)=K$ is called the orbifold fundamental group of $\mathscr O$. In general, the derived subgroup $K'$ of $K$ may have elements acting with fixed points; i.e., it may happen that the homology cover $M_\mathscr O=X/K'$ of $\mathscr O$ is not a geometric manifold: it may have geometric singular points. We are concerned with the problem of deciding when $K'$ acts freely on $X$; i.e., when the homology cover $M_\mathscr O$ is a geometric manifold. In the case $d=2$ a complete answer is due to C. Maclachlan. In this paper we provide necessary and sufficient conditions for the derived subgroup $\mathscr O$ to act freely in the case $d=3$ under the assumption that the underlying topological space of the orbifold $K'$ is the 3-sphere $S^3$.