 Three-branch-point regular surface cover

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.