Suppose that we have an orbifold, for example $S^2(3,5,7)$. There are many references that this is a good orbifold and so finitely covered by a surface. By Riemann-Hurwitz, the surface would have Euler number as multiple of -34.
My question is, does the surface of Euler number exactly -34 cover the orbifold?
More generally, if $k\chi(O)=2m$, is there a surface of Euler number $2m$ covering $O$?
I suspect that it was known earlier, but you can apply a theorem of Edmonds, Ewing and Kulkarni to determine the existence of torsion-free subgroups of the given index in a Fuchsian group, such as $F=\pi_1(S^2(3, 5, 7))$. It follows from their work that $F$ contains a torsion-free subgroup $F'$ of index $k$ if and only if $k$ is divisible by $3\times 5\times 7$. This is equivalent to the existence of a connected surface $S$ which is a $k$-fold cover of your orbifold. Since $$ \chi(S)=k \chi(S^2(3, 5, 7)) = k (\frac{1}{3} + \frac{1}{5} + \frac{1}{7} -1) $$ the problem reduces to simple arithmetic which I will leave to you to work out.
[1] Allan L. Edmonds, John H. Ewing, and Ravi S. Kulkarni, Torsion free subgroups of Fuchsian groups and tessellations of surfaces, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 456-458.
If I remember it correctly, a full proof of their theorem appeared in Inventiones Mathematicae in 1984 or so.
