= Solution
Take $T$ to be a finite group acting by orientation-preserving <isometries>. We give both an unbranched construction for every such abstract group and the orbifold construction needed for low-genus examples. The inherited quotient metric and the existence of some unrelated smooth hyperbolic metric on the underlying topological surface are different questions.
For a free action, choose generators $t_1,\ldots,t_s$ of $T$ and a closed orientable surface of genus $g_0\geq\max(2,s)$. Its <surface group> has generators $a_j,b_j$ and relation $\prod_j[a_j,b_j]=1$. Sending $a_j$ to the chosen generators, the remaining $a_j$ to one, and all $b_j$ to one defines a surjective homomorphism to $T$. Its kernel determines a connected regular finite cover $M$. Give the base a <Riemannian metric> of constant curvature $-1$ by the <uniformization theorem>, and pull it back to $M$. Its deck group is $T$, acting freely and isometrically. All subgroup quotients are then smooth covers as well.
For the more general construction, prescribe an oriented compact hyperbolic <orbifold> of signature $(g_0;m_1,\ldots,m_r)$, with
$$
\chi_{\mathrm{orb}}=2-2g_0-\sum_{i=1}^r(1-1/m_i)<0.
$$
Its <orbifold fundamental group> has presentation
$$
\Gamma=\left\langle a_j,b_j,c_i\ \middle|\ c_i^{m_i}=1,\ \prod_{j=1}^{g_0}[a_j,b_j]c_1\cdots c_r=1\right\rangle.
$$
Choose an epimorphism $\theta:\Gamma\to T$ for which every $\theta(c_i)$ has exact order $m_i$. All finite-order elements of the uniformizing group are conjugate to powers of cone generators, so exact orders make $K=\ker\theta$ torsion-free. Realize $\Gamma$ as a cocompact <Fuchsian group> on the <hyperbolic plane>. Then
$$
M=K\backslash\mathbb H^2,\qquad M_0=T\backslash M=\Gamma\backslash\mathbb H^2,\qquad M_1=U\backslash M=\theta^{-1}(U)\backslash\mathbb H^2.
$$
The source notation $T/M$ is interpreted as the quotient of $M$ by $T$, as in the consistent subgroup notation $U\backslash M$. The manifold $M$ is smooth; $M_0$ has cone angles $2\pi/m_i$ and may have singularities.
The <smoothness criterion from cone-monodromy cycles> is explicit. On the $n=[T:U]$ cosets, let $\sigma_i$ be the permutation induced by $\theta(c_i)$. A cycle of length $\ell$ has local degree $\ell$ over the cone point. The resulting total angle is $2\pi\ell/m_i$, so its residual cone order is $m_i/\ell$. Therefore the inherited metric on $M_1$ is smooth precisely when every cycle of every $\sigma_i$ has length $m_i$. Equivalently,
$$
U\cap t\langle\theta(c_i)\rangle t^{-1}=\{1\}\qquad\text{for every }t\in T\text{ and every }i.
$$
Indeed a nonidentity cone-generator power fixes a coset exactly when its conjugate lies in $U$. The inherited metric on $M_0$ itself is smooth precisely when there are no nontrivial cone stabilizers, equivalently when the $T$ action on $M$ is free.
For the <Euler characteristics>, first distinguish the ordinary characteristic of the underlying base from its <orbifold Euler characteristic>:
$$
\boxed{\chi(|M_0|)=2-2g_0,\qquad\chi_{\mathrm{orb}}(M_0)=2-2g_0-r+\sum_i\frac1{m_i}.}
$$
To derive the covering formulas, triangulate the underlying base with all cone points as vertices. Ordinary vertices, open edges and faces lift with their full degree. A cone point whose monodromy has $c_i$ cycles has $c_i$ preimage vertices rather than $n$; thus the lifted Euler characteristic loses $n-c_i$ there. This is the <Euler characteristic from coset cycle counts>:
$$
\chi(|M_1|)=n\chi(|M_0|)-\sum_i(n-c_i)=n(2-2g_0-r)+\sum_i c_i.
$$
For the full regular cover $M$, each cone generator acts on $T$ in cycles of length $m_i$, so it has $|T|/m_i$ cycles. Consequently
$$
\boxed{\chi(M)=|T|\left(2-2g_0-r+\sum_i\frac1{m_i}\right).}
$$
When $M_1$ is smooth, all its cone cycles likewise have full length and $c_i=n/m_i$. Hence
$$
\boxed{\chi(M_1)=[T:U]\left(2-2g_0-r+\sum_i\frac1{m_i}\right)=\frac{\chi(M)}{|U|}.}
$$
These are also the <Riemann-Hurwitz formula> with the ramification deficits made explicit. The <Gauss-Bonnet theorem> gives area $-2\pi\chi$ for each smooth closed hyperbolic surface, consistently multiplying with covering degree.
Finally, if one asks whether the underlying surface admits any smooth metric of constant curvature $-1$, rather than whether this quotient metric is smooth, the criterion is ordinary <Euler characteristic> less than zero. Necessity follows from the <Gauss-Bonnet theorem>; sufficiency is the <uniformization theorem> for genus at least two. Thus an orbifold quotient with cone singularities can sometimes be given a different smooth hyperbolic metric, but this change does not preserve its role as the locally isometric quotient in the spectral construction. A sphere or torus underlying $M_0$ cannot be made a closed smooth curvature-$-1$ surface at all.
Back to article page
