I'm just getting starting looking into triangle groups for an undergraduate project and had a question about triangle groups generated by reflections in the triangle sides.
The triangles take angles $\pi/a, \pi/b, \pi/c$, in the Euclidean case $\pi/a + \pi/b + \pi/c = \pi$. My reference source says $ a,b,c \in \mathbb{N}$ - why is this so? Could we not also have groups where $a,b,c$ are not all natural numbers?
Edit: The triangle group is a reflection group with presentation $\Delta(a,b,c) = \langle p,q,r \mid p^{2} = q^{2} = r^{2} = (pq)^{c} = (qr)^{a} = (rp)^{b} = 1 \rangle$.
In general, if you, say, work in the Euclidean plane, there will be triangles with arbitrary angles $\alpha, \beta, \gamma$, as long as they satisfy $\alpha+\beta+\gamma=\pi$. Hence, in order to construct such a triangle $T$ you do not need the assumption $a, b, c\in {\mathbb N}$. However, on the next step, you will be taking a group of Euclidean isometries generated by reflections $p, q, r$ in the sides of $T$. What would it even mean for this group to have a presentation with $(pq)^c=1$, etc, if $c$ is not a natural number? For the subgroup to have the presentation that you wrote, one needs $a, b, c$ to be natural numbers. Once you understand the proof, you will see where this condition is used.