Orbifold fundamental group (${\pi }_1^{\mathrm{orb}}(B)$)

The orbifold fundamental group records loops together with their local isotropy relations, so a loop around a cone point of order $m$ has its $m$th power trivial rather than being trivial itself. For an oriented closed surface of genus $g$ with cone orders $m_1,\dots ,m_r$, its presentation is $⟨a_1,b_1,\dots ,a_g,b_g,c_1,\dots ,c_r\mid c_i^{m_i}=1,{\prod }_j[a_j,b_j]c_1\cdots c_r=1⟩$. A torsion-free finite-index subgroup of a hyperbolic surface orbifold gives a smooth locally isometric surface cover. The absence of torsion means that the subgroup meets no nonidentity local cone stabilizer.

Pair the $2s$ edges of an oriented hyperbolic polygon by length-preserving identifications reversing their boundary directions. Label the pairs by generating elements $A_1,\dots ,A_s$ of a finite group $T$. Glue the corresponding edge of a copy labelled $g$ to its mate in the copy labelled $g{A}_k$. Left multiplication of labels defines the $T$ action. A vertex cycle of a paired polygon has angle sum ${\theta }_j$ and accumulated gluing word $C_j\in T$ of order $q_j$; above it lie $|T|/q_j$ vertices, each of cone angle $q_j{\theta }_j$. The action is free away from vertices, and the vertex stabilizers are conjugates of $⟨C_j⟩$. If the base cone has integer order $m_j=2\pi /{\theta }_j$ and $q_j\mid m_j$, this is an orbifold covering, with lifted cone order $m_j/q_j$. The lift is smooth exactly when $q_j=m_j$.

In the paired-polygon construction of finite surface covers, a subgroup $U$ acts freely precisely when the displayed intersection is trivial for every vertex word $C_j$ and every $g\in T$. The quotient map $M\to U\backslash M$ is then a normal covering map with deck group $U$ and $\chi (U\backslash M)=\chi (M)/|U|$. Normality of $U$ in $T$ is instead the criterion for the intermediate orbifold cover $U\backslash M\to T\backslash M$ to be normal. These are different covering maps.

For a paired $2s$-gon with $r$ base vertex cycles of a paired polygon, there are one face, $s$ edges and $r$ vertices on the base, so $\chi (M_0)=1-s+r$. The full finite-group construction has $|T|$ faces, $s|T|$ edges and ${\sum }_j|T|/q_j$ vertices, giving the displayed ordinary Euler characteristic. If all lifted angles are $2\pi$, the orbifold Euler characteristic formula gives equivalently $\chi (M)=|T|{\chi }_{\mathrm{orb}}(M_0)$.

A vertex cycle of a paired polygon is an equivalence class of vertices under the edge identifications. Its angle sum is the total angle at the resulting point of the quotient surface. Recording the edge labels encountered around that point gives a group word, whose order determines the number of turns required to close a lifted cycle. This explains the angle multiplier and lifted vertex count in the paired-polygon construction of finite surface covers.

Over a cone point of order $m$, a cycle of length $\ell$ in the monodromy on sheets has ramification degree $\ell$ and residual cone angle $2\pi \ell /m$. The lifted inherited hyperbolic metric is smooth precisely when every such cycle has length $m$. Equivalently the subgroup defining the cover meets no conjugate of a nonidentity power of a cone generator.