Positive and negative Euler characteristic

A symmetry pattern on the sphere always gives rise to a quotient orbifold with positive Euler characteristic. In fact, if the order of symmetry is , then the Euler characteristic of the quotient orbifold is , since the Euler characteristic of the sphere is 2.

However, the converse is not true. Not every collection of parts costing less than can be put together to make a viable pattern for symmetry on the sphere. Fortunately, the experts at The Orbifold Shop know the four bad configurations which are too skimpy to be viable:

• A single cone point, with no other part, is bad.
• Two cone points, with no other parts, is a bad configuration unless they have the same order.
• A mirror with a single corner reflector, and no other parts, is bad.
• A mirror with only two corner reflectors, and no other parts, is bad unless they have the same order.

All other configurations are good. If they form an orbifold with positive orbifold Euler characteristic, they come from a pattern of symmetry on the sphere.

The situation for negative orbifold Euler characteristic is straightforward, but we will not prove it:

Theorem. Every orbifold with negative orbifold Euler characteristic comes from a pattern of symmetry in the hyperbolic plane with bounded fundamental domain. Every pattern of symmetry in the hyperbolic plane with compact fundamental domain gives rise to a quotient orbifold with negative orbifold Euler characteristic.

Since you can spend as much as you want, there are an infinite number of these.