1.8. The Modular Group and the Veech Group 15

closely related to the subject of translation surfaces. Once again, let

us consider the square torus. A linear transformation of the form

(1.2)

T (x, y) = (ax + by, cx + dy), a, b, c, d ∈ Z, ad − bc = 1.

acts as transformation of the square torus, via the following 4-step

process:

(1) Start with a point p in the square torus.

(2) Choose a point (x, y) such that p represents the collection

of points glued to (x, y).

(3) Subtract off integer coordinates of T (x, y) until the result

(x , y ) lies in the unit square.

(4) The image of the map is p , the point that names the col-

lection of points glued to (x , y ).

Any ambiguity in the process that takes us from p to p is absorbed

by the gluing process.

So, any integer 2 × 2 matrix with determinant 1 gives rise to

a transformation of the square torus that, on small scales, is indis-

tinguishable from a linear transformation. The set of all such maps

forms a group known as modular group. The maps in equation (1.2)

have the same form as the M¨ obius transformations discussed above.

Interpreting the maps in equation (1.2) as M¨ obius transformations in-

stead of linear transformations, we can interpret the modular group

as a group of symmetries of the hyperbolic plane.

The modular group is an object of great significance in mathe-

matics, and we cannot resist exploring some of its properties that are

not, strictly speaking, directly related to surfaces. For instance, in

Chapter 19 we will discuss continued fractions and their connection

to the modular group and hyperbolic geometry. In Chapter 22 we will

see that the modular group is the main ingredient in the proof of the

Banach–Tarski Theorem. The Banach–Tarski Theorem says in par-

ticular that, assuming the axiom of choice, one can cut the unit ball

in

R3

into finitely many pieces and rearrange these pieces so that they

make a solid ball of radius 100000. Though this result seems a bit far

removed from the theory of surfaces, it is quite beautiful and it shows

how objects such as the modular group pop up all over mathematics.