= Three-branch-point regular surface cover
{title2=$\chi(N)=|T|(1/m+1/n+1/r-1)$}
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>.
Back to article page
