TITLE: Periodic billiard paths in hyperbolic triangles
QUESTION [10 upvotes]: It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path,
one obeying the reflection law that the angle of incidence equals the angle of reflection. A rational triangle has angles that are rational multiples of $\pi$.
It remains an open question of whether all triangles have a periodic billiard path.

Q. Is there some analogous theorem for hyperbolic triangles in the hyperbolic plane?

A billiard path in a hyperbolic triangle consists of geodesics which reflect
from the sides of the triangle by the same reflection law. Here is a possible start of
a billiard path inside a triangle drawn in the Poincare half-plane model,
where the geodesics are circular arcs and vertical line segments:

          


Is there some analog of the Euclidean rational angle condition that yields a theorem for hyperbolic triangles?

REPLY [4 votes]: If the triangle is a fundamental domain for a discrete group, a periodic orbit corresponds to a closed geodesic in the quotient surface, which gives you boat-loads of periodic orbits (to be precise, $\sim \exp(L)/L$ of them of length bounded above by $L.$ I think that if all the angles are rational, the triangle is a fundamental domain for a cone-metric, and while a lot of the closed geodesics will go through the vertices, an entropy computation (that is, the set of geodesics passing through the vertices have smaller entropy than the entire geodesic flow) will give you a similar result.