= Branched surface action from two generators
{title2=$\chi(M)=|T|\left(\frac1{|a|}+\frac1{|b|}+\frac1{|ab|}-1\right)$}
For a <finite group> generated by $a,b$, take the regular covering of a thrice-punctured <sphere> with <monodromy> $a,b,(ab)^{-1}$ and fill each lifted puncture by a disk. Locally the projection is $z\mapsto z^m$, where $m$ is the order of its monodromy. The result is a closed oriented <topological surface> with an orientation-preserving group action. The nontrivial <stabiliser subgroups> are the conjugates of $\langle a\rangle$, $\langle b\rangle$ and $\langle ab\rangle$ at the three types of ramification points. Counting lifted cells gives the displayed <Euler characteristic>. A subgroup acts freely exactly when it meets every conjugate of these three cyclic groups trivially. Its quotient is then a <regular covering> of degree equal to the subgroup order, and quotient <Euler characteristic> is divided by that order.
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