Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find it is very restrictive condition. What are other possible values of the inner angles of P for it to be a fundamental domain? For what values it can never be a fundamental domain.?

Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find the angle sum condition in the theorem very restrictive. Is there less restrictive condition on angle sum for P to be a fundamental domain? For example, $P_1$ with angle sum $\pi/2+ \pi/2 + \pi/2+ \pi/3=11 \pi/6$ is not of the form $2\pi/n$ for any natural number $n$, But $P_1$ is a fundamental domain for Fuchsian group generated by reflections on edges of $P_1$.

Let P be a hyperbolic rectangle. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find it is very restrictive condition. What are other possible values of the inner angles of P for it to be a fundamental domain? For what values it can never be a fundamental domain.?

Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find it is very restrictive condition. What are other possible values of the inner angles of P for it to be a fundamental domain? For what values it can never be a fundamental domain.?