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The orbifold fundamental group is the group of deck transformations of the universal orbifold cover. It retains the local finite group data that the ordinary fundamental group of the underlying topological space forgets.^[ref]

For an oriented closed topological surface of genus $g$ with cone points of orders $m_1,\dots ,m_r$, the group presentation isEach local cone relation says $c_i^{m_i}=1$; it does not guarantee that $c_i$ has order exactly $m_i$ in the resulting global group. For the teardrop orbifold, a two-sphere with one cone point of order $m>1$, the remaining relation is $c_1=1$, so its orbifold fundamental group is trivial despite nontrivial local isotropy.^[ref]

For a complete hyperbolic surface orbifold, the universal cover is the hyperbolic plane. A torsion-free finite-index subgroup then acts freely and gives a smooth surface cover. The hyperbolic covering model is what identifies local cone stabilizers with actual finite subgroups in this case.

A teardrop orbifold is an orbifold whose underlying topological space is a two-sphere and which has exactly one cone point of an orbifold of order $m>1$. A neighborhood of that point is modeled on a disk modulo rotations by multiples of $2\pi /m$; every other point has trivial local group.

Topologically, a disk modulo this finite rotation group is still a disk. The local stabilizer is therefore extra orbifold information invisible in the underlying sphere. The singularity order distinguishes teardrops that have the same underlying topological space.

The teardrop is a standard example of a bad orbifold: it has no manifold orbifold covering. Cutting it into an ordinary disk and a disk with one order-$m$ cone point exposes incompatible nontrivial covering behavior along their common boundary. Thus being locally a finite quotient does not imply being globally covered by a manifold.^[ref]