Introduction

In the study of geometric structures and complex analysis, the notion of a fundamental domain arises naturally when we consider the action of a group on a space. Fundamental domains serve as building blocks, allowing us to understand complicated spaces by analyzing a single, representative piece and its images under the group action.

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1. Definition

Let $X$ be a topological space, (often a manifold or a complex domain) and let $\Gamma$ be a group acting on $X$. The group $\Gamma$ is typically discrete (such as a lattice of translations in $\mathbb R^n$ or a Fuchsian group acting on the upper half-plane).

A fundamental domain for the action of $\Gamma$ on $X$ is a subset $D\subset X$ satisfying the following conditions:

1. Covering / Exhaustion:
   Every point in $X$ lies in the orbit of some point in $D$, that is, $X=\bigcup\limits_{g\in \Gamma}g\cdot D$.
2. Non-overlapping / Disjoint Interiors:
   The interiors of distinct translates are disjoint: $g_1\neq g_2\implies g_1\cdot Int(D)\cap g_2\cdot Int(D)=\phi$.

Informally, a fundamental domain is a single tile of the space $X$ under the action of $\Gamma$. The group action reproduces the entire space by translating, rotating, or otherwise transforming this tile, without overlapping interiors.

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2. Geometric and Complex-Analytic Intuition

Fundamental domains are closely linked to the concept of quotient spaces. If $D$ is a fundamental domain for the action of $\Gamma$ on $X$, then the quotient space $X / \Gamma$ can be constructed by identifying points on the boundary of D according to the group action.

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3. Examples

Example 1: Lattices in $\mathbb R^2$

Consider the lattice $\Gamma=\mathbb Z\oplus\mathbb Z$ acting on $\mathbb R^2$ by translations:$(m,n)\cdot (x,y)=(x+m,y+n), \text{ for all }(m,n)\in \mathbb Z^2$

A parallelogram spanned by two lattice vectors serves as a fundamental domain. Translating this parallelogram along all lattice vectors tiles the entire plane without overlapping interiors. The quotient $\mathbb R^2/\mathbb Z^2$ is naturally homeomorphic to the 2-torus $\mathbb T^2$.

Moreover for different choices of lattice bases $\{\omega_1,\omega_2\}$, the action of the lattice $\Gamma_{\omega_1,\omega_2}=\mathbb Z\cdot\omega_1+\mathbb Z\cdot \omega_2$ on $\mathbb R^2$ or $\mathbb C$, might produce different, conformally non-equivalent tori (all are homeomorphic but not biholomorphic), with the distinct conformal equivalence classes determined by the complex number $\frac{\omega_1}{\omega_2}\in \mathbb C$. When put together, the collection of all these biholomorphic classes form the so called Moduli space of 2-torus.

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Example 2: $\mathbb Z$ action on $\mathbb R^2$ by translation, producing a cylinder

Let $\Gamma=\mathbb Z$ act on $\mathbb R^2$ by integer translations along the x-axis: $n\cdot (x,y)=(x+n,y),\text{ for all }n\in \mathbb Z$.

A vertical strip of width 1, say $D=\{(x,y)\in \mathbb R^2: 0\leq x\leq1\}$,

serves as a fundamental domain. Its translates tile the plane horizontally. The quotient $\mathbb R^2/\mathbb Z$ is a cylinder, which is homeomorphic (in fact biholomorphic) to the punctured complex plane $\mathbb C^*=\mathbb C\setminus \{0\}$ via the exponential map $w = e^{2\pi i z}$.

The following video does a great job with the visualization of the equivalence between the infinite cylinder and the punctured plane.

Example 3: $\mathbb Z$ action on the upper half-plane $\mathbb H$ producing a punctured disc

Let $\Gamma=\mathbb Z$ act on the upper half-plane $\mathbb H :=\{z\in \mathbb C: Im(z)>0\}$ by unit translations:$n\cdot z=n+z,\text{ for all }n\in \mathbb Z$.

A vertical semi-infinite strip $D=\{z\in \mathbb H: 0\leq z\leq 1\}$ is a fundamental domain. Its images under $\Gamma$ tile the entire upper half-plane. The quotient $\mathbb H/\mathbb Z$ is biholomorphic to the punctured unit disc $\mathbb D^*=\mathbb D\setminus \{0\}=\{z\in \mathbb C: 0<|z|<1\}$ via the exponential map $w = e^{2\pi i z}$.

This example illustrates how simple translations produce nontrivial Riemann surfaces.

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Example 4: Modular group on the upper half-plane

Let $\Gamma=SL(2,\mathbb Z)$ act on $\mathbb H$ via Möbius transformations:$\gamma\cdot z=\frac{az+b}{cz+d}, \text{ where }\gamma=\begin{pmatrix}a & b\\c&d\\\end{pmatrix}\in SL(2,\mathbb Z)$.

A classical fundamental domain is$D=\{z\in \mathbb H: |z|\geq1\text{ and }−\frac{1}{2}​\leq Re(z) \leq\frac{1}{2}\}$.

Its images tile the upper half-plane. This domain is central in modular form theory and in constructing quotient surfaces of genus zero with cusps.

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4. Analytical and Topological Perspectives

• Analytical viewpoint: Functions invariant under the group action can be studied by restricting them to a fundamental domain. For example, periodic functions on $\mathbb R^2$ are determined by their values on a single parallelogram of the lattice. Modular forms are determined by their values on the modular fundamental domain.

• Topological viewpoint: Fundamental domains often have identifiable boundaries, and identifying these boundaries via group actions constructs the quotient space $X / \Gamma$. For instance, a vertical strip under $\mathbb Z$ action yields a cylinder or punctured disc; a parallelogram under $\mathbb Z^2$ yields a torus. The knowledge of fundamental domain makes it easy to identify a quotient space.

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5. Closing Remarks

Fundamental domains illustrate how symmetry organizes space. They connect:

• Geometry: tilings, tessellations, and CMC surfaces.

• Topology: quotient spaces and surface classification.

• Complex analysis: Riemann surfaces, modular forms, and analytic continuation.

From lattices and strips to Möbius transformations, fundamental domains provide a single window into infinite, symmetric structures. They allow us to reduce global complexity to a local study, a principle that echoes throughout mathematics.