Dirichlet region

Description

Der Dirichlet Bereich definiert durch Angabe eines Punktes einen natürlichen Fundamentalbereich. Die Idee ist hierbei, von jedem Punkt des Orbits ein Voronoi-Diagramm expandieren zu lassen. Die Zellen bilden dann einen fundamentalen Bereich.

The Dirichlet Region has some nice properties. The boundary between two regions is always a geodesing running perpendicular to the geodesic connecting the two centres.

Definition

Sei $\Gamma$ eine Fuchsian group und sei $p\in H$ ein Punkt, der von keinem nichttrivialen Element von $\Gamma$ fixiert wird. (Solche Punkte existieren immer) Der Dirichtlet Bereich ist definiert durch $D_p(\Gamma ):=\{z\in H:d(z,p)\le d(z,\Gamma p)\}$

Properties

Fundamental Region

If $p$ is not fixed by any nonidentity-element of the Fuchsian group $\Gamma$, then $D_p(\Gamma )$ is a connected fundamental region for $\Gamma$.

Convexity

The Dirichlet Region is the intersection of hyperbolic half-planes and therefore convex in hyperbolic space.

Shape

The shape of Dirichlet Region can be quite complicated. The only real restrictions are that they need to be convex in hyperbolic space and be bounded by geodesics of part of the real axis. The shape of the fundamental region reflects the group itself. (But not in an entirely trivial way) I don’t think the first image can happen in a subgroup of $\Gamma =SL(2,\mathbb{Z})$ since its fundamental region is a triangle with two points in infinity and subgroups of $\Gamma$ must have bigger fundamental groups.

Two sides of a region are called congruent if there is a transformation moving one side to the other. This transformation is unique.

Congruent sides pairs side of a Dirichlet region generate the group

Let $\{T_i\}$ be the subset of $\Gamma$ consisting of those elements which pair the sides of some Dirichlet region $F$. Then $\{T_i\}$ is a set of generators for $\Gamma$.

Proof: It is known, that the Dirichlet tessellation is “well-connected” meaning every region can be reached by crossing finitely many regions in between. Let $R$ be the starting region. Every region corresponds to an element $T\in \Gamma$. The Elements $\{T_i\}$ allow us to move $TR$ to $T$ by induction.