Definition 2.1. A triangle group, denoted by three parameters $(\alpha, \beta, \gamma)$ such that $2 \leq \alpha, \beta, \gamma$, is generated by the reflections over the edges of a triangle with angles $\pi / \alpha, \pi / \beta$, and $\pi / \gamma$. The group rule is composition of reflections.
Note that if $\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=1$, the triangle sits in Euclidean space. If the sum is greater than 1 , it sits in elliptic space and if it is less than 1 , it sits in hyperbolic space. We will focus on this third case because the first two are fairly limited (there are only three examples in Euclidean space). If we let the generators of our group be $a, b, c$ where $a$ is the reflection over the edge opposite angle $\pi / \alpha$ etc., we have the following relations:
\[
a^2=b^2=c^2=1,(a b)^\gamma=(b c)^\alpha=(a c)^\beta=1
\]
The first relation holds because applying any reflection twice is the identity, and the second relation holds because by reflecting over each edge incident to an angle repeatedly, we can rotate around that angle fully, again yielding the identity. Now, in the Klein Model of the hyperbolic plane, each reflection can be represented by a projective transformation, so we also have a corresponding group of matrices $A \in M_{3,3}$