Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are all integral submultiples of $\pi$, then the group generated is discrete and has a presentation as a Coxeter group.

**Questions:**

What other choices of angles generate a discrete group? For instance, what if the angles of the triangle are just rational multiples of $\pi$?

*As an example, reflections in the Euclidean triangle with angles $\frac{\pi}{6}, \frac{\pi}{6}, \frac{2\pi}{3}$ generate the (discrete) $(2,3,6)$ triangle group.*When the group that is generated is discrete, can one easily find a fundamental domain for the action, e.g. in terms of the angles of the original triangle? Is the fundamental domain necessarily also a triangle?

For discrete groups generated in the hyperbolic plane, which choices of angles yield arithmetic subgroups of $\operatorname{PSL}(2,\mathbb R)$?

*If the discrete groups generated are themselves triangle groups, then Takeuchi classifies which are arithmetic.*