= Paired-polygon construction of finite surface covers
Pair the $2s$ edges of an oriented <hyperbolic polygon> by length-preserving identifications reversing their <boundary> directions. Label the pairs by generating elements $A_1,\ldots,A_s$ of a finite <group> $T$. Glue the corresponding edge of a copy labelled $g$ to its mate in the copy labelled $gA_k$. 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 $\langle C_j\rangle$. 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 $q_j=m_j$.
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