Course MT3818 Topics in Geometry

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## Isometries of the line

The group I(R) of isometries of the line R is an interesting and (for such an apparently easy case) complicated group.

Since the group O(1) = {±1} we can divide the elements of the group into two subsets: those with orthogonal part +1 and those with the orthogonal part -1. The former are just translations of the form xa + x while the latter are maps of the form xa - x.

These latter maps have the effect of "reversing the direction" on the line. To see more clearly what they do, observe that the point -a/2 is mapped to itself and so you can think of these maps as reflection in this fixed point.
Reflection in the point b is the map x ↦ 2b - x.

Observe that composing two of these reflections gives a translation. Calculation should show you that composing reflections in points b and c gives translation by twice the distace between b and c. The direction of the translation depends on the order in which you do the reflections and so the group is non-abelian.

Notice that translation by any a ≠ 0 has infinite order, while all the reflections have order 2.

We can think about this group I(R) in another way. Since the linear part LO(1) = {±1} we can represent the element TaL by the pair (a, ±1).

This gives I(R) = {(a, b) ∈ R2 | b = ±1 } and composing the maps gives a multiplication * on these pairs defined by

(a1 , b1)*(a2 , b2) = (a1 + b1a2 , b1b2).

It is now no longer obvious that the multiplication is associative. The identity element (which is, of course, the pair representing the map xx) is the pair (0,1). The inverse of an element (a, 1) is (-a, 1) (translation in the opposite direction) while an element (a, -1) is its own inverse.

The group of matrices { | a, bR, b = ±1 } is isomorphic to I(R). See Exercises 3 Question 3.

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JOC March 2003