Why are Fuchsian groups interesting?

Question

I am recently reading the book "Fuchsian groups" by Katok and now on Chapter $2$. I am curious about why Fuchsian groups are interesting. I look it up online and find answers here. Those are great answers. However, I am at an undergrad level now and I don't get a very clear picture. Hyperbolic geometry is interesting since there are some great properties, like hyperbolic Gauss-Bonnet. Would someone like to provide an example, e.g. rubric cube when studying group theory?

Appreciate any resources and helps!

Addendum: Fuchsian groups are discrete subgroups of PSL($2$, $\mathbb{R}$) or automorphism of the upper-half plane.

Answer 1

Well, there might be other answers, but I think a very clear motivation is the study of surfaces.

Namely, consider a closed orientable surface of genus $1$, i.e. a torus. You probably know that the torus can be obtained by taking a square and identifying its opposite sides adequately. Somewhat more precisely, we can pair the sides of a square via Euclidean isometries to obtain a torus.

Naturally, we want to know whether this can be done for surface of higher genus (with more holes), and it turns out that hyperbolic polygons provide the answer. As you will learn in Katok's book, not every side-pairing of a polygon will work (for example, at the end of day you'd like your surface to be closed, and this imposes a restriction on the sum of the angles of your polygon).

Going back to hyperbolic polygons, consider (for simplicity) the closed orientable surface of genus $2$. It is not so hard to see that you may glue the sides of an (hyperbolic) octagon to obtain such a surface. Now, how do we describe this side-pairing? In the case of the torus, this is achieved via translations (which are discrete Euclidean isometries), and it occurs that for hyperbolic polygons, this side-pairing is achieved precisely by discrete subgroups of the isometry group of the hyperbolic plane; in other words, Fuchsian groups!

This means that understanding Fuchsian groups yields an understanding of compact orientable surfaces, although I must say that Fuchsian groups are interesting in their own right : )

Answer 2

Fuchsian groups appear in several areas of mathematics, including complex analysis, number theory, topology. Number theorists really like Fuchsian groups like $PSL(2, {\mathbb Z})$ (and its relatives) and many deep number-theoretic questions can phrased in terms of such groups. For instance, Shimura-Taniyama Conjecture (proven by Andrew Wiles), which, in turn, implies Fermat's Last Theorem, can be formulated in terms of certain subgroups of $PSL(2, {\mathbb Z})$ called congruence subgroups. I can provide more details on this in case you are interested.