16 1. Book Overview
Getting back to translation surfaces, we will see in Chapter 18
that one can associate to any translation surface a group of symme-
tries of the hyperbolic plane. This group is known as the Veech group
of the translation surface. It often happens that the Veech group is
trivial, or very small, but for many special examples the Veech group
is large and beautiful. For instance, the Veech group associated to the
regular octagon surface is closely related to a tiling of the hyperbolic
plane by triangles having angles 0, 0, and π/8. One of the highlights
of Chapter 18 is a discussion of (essentially) this example.
1.9. Moduli Space
The square torus is not the only translation surface without any cone
points. In Chapter 20 we consider the family M unit area parallelo-
gram, in the same pattern as in Figure 1.1. Essentially the same anal-
ysis we made in connection with Figure 1.1 can be made in connection
with any surface in our family. All these surfaces are seamlessly flat
at each point. A myopic bug on any of these surfaces would not be
able to tell that he was not in the Euclidean plane.
On the other hand, these various surfaces are typically not the
same geometrically. For instance, a surface made from a long thin
rectangle obviously has diameter greater than the diameter of the
square torus. Similarly, such a surface has a very short essential loop
whereas all essential loops on the square torus have length at least 1.
We can consider the family M as a space in its own right. Each point
of M corresponds to a different flat torus. This space M is known as
the moduli space of flat tori. We will discuss M and related objects
in Chapter 20.
Amazingly, M turns out to be a surface in its own right, and
(with the exception of 2 special points) this surface is modelled on
hyperbolic geometry! Just to repeat: the space of all tori made by
gluing together unit area parallelograms turns out to be a surface that
naturally wears hyperbolic geometry (away from 2 special points).
One of the special points in M corresponds to the square torus, and
the other one corresponds to the surface obtained by gluing together
the opposite sides of a rhombus made from 2 equilateral triangles.
Referring to the discussion of covering spaces above, we can consider