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Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller than $\pi/2$?

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up vote 7 down vote accepted

Yes. One can extend the Gauss-Bonnet theorem to 2-orbifolds since but area an orbifold Euler characteristic behave well under the covering map.

To find all hyperbolic 2-orbifolds with area smaller than $\pi/2$, we need to get a list of orbifolds $\{O_i\}$ $0>\chi(O_i)>-1/4$.

Thankfully, this is not a horrible task. Background and pictures for this computation is in Chapter 13 of Thurston's notes, although there are certainly older sources. To set notation, $O$ is our orbifold, $X_O$ is the underlying topological space of this orbifold. Also, $O$ can have three types of singularities cone points of order $n_j$ (points fixed locally by rotations of order $n_j$ and not any reflections), points fixed by reflections but not any rotations, and corner reflections (points fixed locally by a dihedral group). Note, fixed by reflections but not fixed by a dihedral group will not contribute to this computation and if a point at infinity is fixed at least to reflections we say the dihedral group that fixes it is infinite. If an orbifold $O$ has $n$ cone points each fixed by a cyclic group of order $r_i$ and $m$ corner points each fixed by a dihedral group of order $t_j$ then the orbifold Euler characteristic is:

$$\chi(O)= \chi(X_O) - \Sigma_{i=1} ^n (1-1/r_i) -\Sigma_{j=1} ^m (1/2-1/t_j) $$

(if $t_j$ is $\infty$ of course $1/t_j$ should be treated as $0$).

Now in order to get a orbifold $O$ such that $0>\chi(O_i)>-1/4$. The underlying space has to be $S^2,D^2,RP^2$.

Also, adapting the same notation as the reference say $Q(r_1,...,r_n;t_1,..,t_m)$ is the orbifold with base space $Q$ cone points of orders $r_i$ and corner points of orders $t_j$.

If there are corner points, then the underlying space $X_O$ must be $D^2$. Each corner point contributes $-(t_j-2)/(2t_j)\leq -1/4$ to the Euler characteristic, and so there are at most 4 of them.

Each cone point contributes $-(r_i-1)/r_i\leq -1/2$ to Euler characteristic. So there are at most 2 cone points if the underlying space is $D^2$ or $RP^2$ and at most 4 if the base space is $S^2$.

From that, one should be able to make a list.

Note: If one allows equality, $\chi(O)=-1/4$, then an orbifold with underlying space an annulus or Mobius band and one corner point of order 4 should also be included on the list. As well as the orbifold with underlying space a disk and 5 corner points of order 4. There are few others as well, but they all have underlying space $S^2,D^2$ or $RP^2$.

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Thanks very much! – Mikhail Katz Dec 5 '13 at 13:12

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