I am having a hard time finding references(apart from wikipedia) for the geometric interpretation of triangle groups $$T_{a,b,c} =\langle x,y: \, |x|=a,|y|=b,|xy|=c \rangle.$$ How can these groups be visualized in the Euclidean, spherical and hyperbolic cases?
Many thanks in advance!
2 Answers
A good way to visualize them is to construct a triangle whose angles are $\frac{\pi}{a}$, $\frac{\pi}{b}$, and $\frac{\pi}{c}$, then to reflect the triangle across each of its sides to get new triangles, then reflect those triangles across their sides, etc. etc.
"Of course", you may say, "this is not always possible. In fact, it is possible only when $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}=1$." That is a valid response IF you are thinking Euclidean triangles. Using this you can compute exacly which triples of integers $a,b,c$ give triangle reflection groups in the Euclidean plane, for example $(a,b,c)=(2,4,4)$. And there are only finitely many such triples. You can easily list out for yourself, draw each of those triangles, and start flipping them across their sides to visualize the reflection group.
On the other hand if you consider examples where $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} > 1,$ then you can construct triangles like that on the sphere, each of which gives a triangle reflection group in the sphere. Again there are finitely many such triples, for example $(a,b,c)=(4,4,4)$, together with an infinite family of triples of the form $(2,2,c)$ for which corresponding reflection group contains an index 2 subgroup isomorphic to the finite dihedral group $D_{2c}$. These triples can be easily listed out, and the ones not related to dihedral groups can all be visualized on the sphere and are closely related to the five Platonic solids.
All of the infinitely many other triples, the ones satisfying $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1$, are angles of triangles in the hyperbolic plane. The most extreme example, the one which makes the left hand side of the inequality closest to $1$, is $(a,b,c)=(2,3,7)$.
Draw a triangle with angles $\pi/a$, $\pi/b$, and $\pi/c$ in the geometry $H$ where $H$ is spherical, Euclidean, or hyperbolic, accordingly as $\pi/a + \pi/b + \pi/c$ is greater than, equal to, or less than $\pi$. Then $x$ and $y$ can be interpreted as rotations by $2\pi/a$ and $2\pi/b$ about the corresponding vertices $A$ and $B$.
Noneuclidean Tesselations and Their Groups by Wilhelm Magnus is a standard reference. Also, Section 2 of this paper gives an explicit embedding of $x$ and $y$ as matrices in the hyperbolic case. It also has some useful illustrations.
