Understanding Fuchsian Groups

What are Fuchsian Groups?

Fuchsian groups are discrete groups of isometries of the hyperbolic plane. They act on hyperbolic space while preserving distances and angles, creating beautiful tessellations that have captivated mathematicians for over a century.

The Modular Group \(\text{SL}_2(\mathbb{Z})\)

The most important Fuchsian group is \(\text{SL}_2(\mathbb{Z})\), consisting of 2×2 matrices with integer entries and determinant 1:

\[\text{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a,b,c,d \in \mathbb{Z}, \; ad - bc = 1 \right\}\]

Technically, the Fuchsian group is \(\text{PSL}_2(\mathbb{Z}) = \text{SL}_2(\mathbb{Z})/\{\pm I\}\), where we identify each matrix with its negative. This group lies at the heart of modular forms, elliptic curves, and much of modern number theory.

The Generators \(S\) and \(T\)

Remarkably, the entire infinite group is generated by just two matrices:

\[S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \quad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\]

These act as Möbius transformations on the upper half-plane \(\mathbb{H}\):

\[S(z) = -\frac{1}{z} \quad \text{(inversion)} \qquad T(z) = z + 1 \quad \text{(translation)}\]

The Fundamental Domain

The fundamental domain is the iconic hyperbolic triangle with vertices at \(\omega = e^{2\pi i/3}\), \(\bar{\omega} = e^{\pi i/3}\), and the cusp at \(\infty\).

📚 Classic Image: This tessellation appears in "nearly every modern book on number theory, algebra, complex analysis, and Riemann surfaces" — a testament to its fundamental importance.

The Poincaré Disk Model

We visualize hyperbolic geometry in the Poincaré disk, where the entire infinite hyperbolic plane fits inside a unit circle. The conformal map:

\[z \mapsto \frac{z - i}{z + i}\]

transforms the upper half-plane to the disk. In this model, hyperbolic straight lines appear as circular arcs orthogonal to the boundary.

Connections to Number Theory

Modular forms — functions with special transformation properties under the group
Elliptic curves — connected via the modularity theorem
Congruence subgroups like \(\Gamma_0(N)\) — crucial for modular forms of various levels
The congruent number problem — an ancient question about rational triangles

🌀 Fuchsian Groups & The Modular Group