# When are groups generated by reflections in a triangle discrete?

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:

1. 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.

2. 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?

3. 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.

• Coxeter - Discrete groups generated by reflections must be relevant, although I'm not sure if it directly answers your questions 1 and 3. For question 1, Lemma 4.2 is relevant but (as you pointed out on my deleted answer) not decisive. Pp. 595–596 describes the "Fricke–Klein construction for a fundamental region": take a never-fixed point $P$, and consider the region bounded by the perpendicular bisectors of the lines connecting $P$ to its transforms. Aug 12, 2022 at 1:50
• If reflections in a triangle generate a discrete group, then it will be tessellated by the fundamental domain. See: en.wikipedia.org/wiki/… Aug 12, 2022 at 10:05
• Regarding your final comment, I think the resulting discrete group must be a triangle group, since it will have three conjugacy classes of torsion elements and these classes generate.
– HJRW
Aug 12, 2022 at 14:43