gt.geometric topology - Orbifold fundamental group in terms of loops?

There is also a natural interpretation of the orbifold fundamental group in terms of loops, using an extended version of a Wirtinger presentation. Let's start out with the closed case and mention the cusped case at the end.

As a warm-up, consider an orbifold that has singular locus a link with underlying space $S^3$. To compute the fundamental group of the orbifold can be computed first from the Wirtinger presentation of the link and then by introducing torsion relations for each meridian for example if the cone angle along the link is $2pi/3$ everywhere, then each meridian $mu_i$ should have the relation $mu_i^3$. A loop around a link component does not bound a (smooth immersed) disk in the orbifold, instead it bounds a disk with a cone point of order 3. However $mu_i^3$ does bound a disk in the orbifold, and is trivial.

To make this interpretation general, we need to consider 3-orbifolds more generally. A geometric 3-orbifold is really a trivalent graph with edges decorated by torsion (or cone angles, depending on taste) embedded in a 3-manifold, and so the underlying manifold, the embedding, and the trivalent points need to be accounted for.

In reverse order, the trivalent points are introduce relations abc=1 (compare to finite subgroups of SO(3) which are actually the isotropy subgroups that fix these types of points). The next two conditions really have to be considered together. Really what needs to be computed is a Wirtinger presentation of the complement of a trivalent graph in the underlying space. Then quotient out by the torsion relations and relations coming from the trivalent points.

For cusped (geometric) manifolds, there are extra ways to decorate the graph, namely there can be trivalent points where the orders of torsion along edges incident to the these vertices are (2,4,4), (2,3,6) or (3,3,3) and there could be 4-valent points where the torsion orders of the edges are (2,2,2,2).

It should be pointed out that such a computation can also be done by Damian Heard's ORB, which is extremely useful if a large example needs to be considered.