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For a finite group generated by $A,B$, glue sheets labeled by the group over the three-punctured Riemann sphere using left monodromy multiplication. Fill the punctures by the local power charts dictated by the orders $m,n,r$ of $A,B,AB$. Right multiplication by inverses defines the commuting deck action. Its nontrivial stabilizers are conjugates of the corresponding cyclic subgroups. The Riemann-Hurwitz formula gives the displayed Euler characteristic. A compatible metric can be averaged over the group; in genus at least two the invariant hyperbolic metric is canonical by the uniformization theorem.

Triangulate the target with all branch values among its vertices. For a degree-$d$ map, every edge and face has $d$ lifts. Above a target vertex $v$, the number of lifted vertices isbecause ${\sum }_{p\mapsto v}{e}_p=d$. Thuswhich is equivalent to the Riemann-Hurwitz formula.

For a reduced rational map $R=p/q$, its derivative is $W/q^2$ with $W=p^{\prime }q-p{q}^{\prime }$. Away from target-coordinate poles, the zeros of this Wronskian identify its ramification points of a holomorphic map; at a pole use the reciprocal coordinate $q/p$. A pole of order $m$ contributes ramification order $m-1$, and infinity must also be checked in local coordinates. For degree $N$ the Riemann-Hurwitz formula gives total ramification $2N-2$. The affine polynomial $W$ can have fewer finite zeros because some ramification lies at infinity. In the rational map approximation for Skyrmions, these directions have zero angular baryon density.