= Orbifold fundamental group
{title2=$\pi_1^{\rm orb}(B)$}
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.<The Geometry and Topology of Three-Manifolds by William Thurston>[Chapter 13, Definition 13.2.5, page 307]{ref}
For an oriented closed <topological surface> of <genus> $g$ with <cone points> of orders $m_1,\ldots,m_r$, the <group presentation> is
$$
\left\langle a_1,b_1,\ldots,a_g,b_g,c_1,\ldots,c_r\ \middle|\ c_i^{m_i}=1,\quad\prod_{j=1}^g[a_j,b_j]c_1\cdots c_r=1\right\rangle.
$$
Each local cone relation says $c_i^{m_i}=1$; it does \b[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.<The Geometry and Topology of Three-Manifolds by William Thurston>[Chapter 13, Definition 13.2.3 and the teardrop example, page 304]{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.
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