A particularly important function is Klein’s *j*-function. It is defined on the upper half-plane of the complex numbers, and is incredibly symmetrical. For example, it is periodic, and thus invariant under translations by integers:

It is also invariant under a larger group of Möbius transformations, namely those corresponding to matrices with integer entries. This group is called the *modular group*, or , since it is the special linear group of invertible matrices with integer entries, quotiented out by its centre of order 2.

We can think of the upper half-plane as Klein’s model of the hyperbolic plane, in which case the *j*-function is a function on the hyperbolic plane and the Möbius transformations correspond to orientation-preserving isometries. Hence, an equivalent definition of the modular group is the orientation-preserving symmetry group of a regular tiling of the hyperbolic plane: the Voronoi tessellation of the zeros of the *j*-function.

Recall that the group of proper orthochronous Lorentz transformations is isomorphic to . Consequently, we can think of the modular group as a discrete version of this.

### Fundamental groups and the complement of the trefoil knot

A surprising isomorphism pertains to knot theory and homotopy. Let *X* be a path-connected topological space. Then, we consider closed paths in *X*, beginning and ending at some pre-specified point. Under concatenation of paths, these form a monoid. When we quotient out by the relation ‘two paths are equivalent if they can be continuously deformed into each other’, then we get the *fundamental group* of *X*.

The fundamental group of any simply-connected space is just the trivial group, as any two paths can be continuously deformed into each other. The once-punctured plane has the infinite cyclic group as its fundamental group, since a path is determined up to continuous deformation by the number of times it encircles the puncture. The fundamental group of the twice-punctured plane is the free group on two generators, best visualised by its Cayley graph:

Now embed the trefoil knot in and consider its complement. It is a remarkable fact that the fundamental group of the resulting space, when quotiented by its centre, yields a group isomorphic to the modular group!

that’s not the free group on two generators…

What isn’t the free group on two generators? The fundamental group of the twice-punctured plane or [the group corresponding to] that Cayley graph? I’m pretty sure they are both the free group on two generators.

The cayley graph you have isn’t the free group on two generators.

There’s only one 4-regular tree, and this is it…