Hilbert's arithmetic of ends

Hilbert's arithmetic of ends is an algebraic approach introduced by German mathematician David Hilbert for Poincaré disk model of hyperbolic geometry . Hilbert defines a field of ends with a multiplicative distance function over the field. So, one can also set up a hyperbolic analytic geometry and hyperbolic trigonometry, whereby any geometric problem can be translated into an algebraic problem in the field.

In Poincaré model of hyperbolic plane, every limiting parallel ray intersects at a point lying on the limit circle which is not included in the hyperbolic plane. So one may take a line as having uniquely two ends. Thus, an addition and a multiplication over the set of these ends constructs a field .

A line of the geometry is represented by an ordered pair (a,b) of distinct elements of the field of ends. On contrary to cartesian plane of Euclidean geometry, a point is represented by a point equation which is satisfied by every line passing through the point, i.e. a pencil of intersecting lines represents a unique point.

Now, we will define the operations over the set of ends, H.

Addition over ends

Addition defined by Hilbert's arithmetic of ends.

Addition defined by Hilbert's arithmetic of ends.

Definition.

Given two ends $\alpha ,\beta$ not equal to $\infty$ and any point C on the line $(0,\infty )$. Let A be its reflection in the line $(\alpha ,\infty )$. Let B its reflection in the line $(\beta ,\infty )$. Then $\alpha +\beta$ is the end of the perpendicular bisector of AB other than $\infty$.

The addition is well-defined, and makes the set (H,+) an abelian group with additive identity 0.

The addition is usually understood as

$\sigma _{\alpha +\beta }=\sigma _{\beta }\sigma _{0}\sigma _{\alpha }$

where for any end $\alpha$, $\sigma _{\alpha }$ denotes reflection in the line $(\alpha ,\infty )$. Thus the addition is independent of the choice of C.

Theorem of three reflection.

The existence of such an addition operation is based on a theorem of three reflection.

Theorem.

Given three lines a ,b, c in the hyperbolic plane with a common end $\omega$, there exist a fourth line d with end $\omega$ such that reflection in d is equal to the product of the reflections in a, b, c:

$\sigma _{c}\sigma _{b}\sigma _{a}=\sigma _{d}$

where $\sigma _{l}$ denotes the reflection in the line l.

So it is clear that the addition is just taking a, b, c as $\alpha$, 0, $\beta$ and d as $\alpha +\beta$ respectively.

Note that an end is an equivalence class of limiting parallel rays. One can fix a hyperbolic line and label its ends 0 and $\infty$. Here H is the set of all ends in the plane different from $\infty$, then one sets $H'=H\cup {\infty }$, so that H' is the set of all ends of the plane. We will make this set into an abelian field by defining also a multiplication on it.

Multiplication over ends

Multiplication over ends

The multiplication over the field is defined by fixing a line perpendicular to the line $(0,\infty )$ where they meet at the point O, and label one of its ends 1, the other -1.

Definition.

Given ends $\alpha ,\beta$, the lines $(\alpha ,-\alpha )$ and $(\beta ,-\beta )$ meet the line $(0,\infty )$ with right angles at A and B, respectively.

So C at where the line $(\alpha \beta ,-\alpha \beta )$ is perpendicular to $(0,\infty )$, is the point which satisfies the relation,

BA'=OC

where the point A' is the reflection of A respect to the point O.

In other words, the point C satisfies the relation OA+OB=OC, according to the euclidean segment addition. So the field has an additive multiplication over line segments. It makes $(H\setminus \{0\},\cdot )$ an abelian group with identity 1.

Rigid motions

Let $\Pi$ be a hyperbolic plane and H its field of ends, as introduced above. In the plane $\Pi$, we have rigid motions and their effects on ends as follows:

• The reflection in $(0,\infty )$ sends $x\in H'$ to -x.

$x'=-x$.

• The reflection in (1,-1) gives,

$x'={1 \over x}$.

• Translation along $(0,\infty )$ that sends 1 to any $a\in H$, a>0 is represented by

$x'=ax$.

• For any $a\in H$, there is a rigid motion $\sigma _{{1 \over 2}a}\sigma _{0}$, the composition of reflection in the line $(0,\infty )$ and reflection in the line (${1 \over 2}a,\infty )$, which is called rotaion around $\infty$ is given by

$x'=x+a$.

• The rotation around the point O, which sends 0 to any given end $a\in H$, effects as

$x'={\frac {x+a}{1-ax}}$

on ends. The rotation around O sending 0 to $\infty$ gives

$x'=-{1 \over x}$.