Assaf Wool's Home Page

Fundamental Domains for Congruence Subgroups of PSL₂(Z)

I have compiled a list of fundamental domains for congruence subgroups of PSL₂(Z) of type &Gamma₀(N) and type &Gamma₁(N) acting on the complex upper half plane. Each subgroup entry contains a description of the domain and a list of generators that correspond to the domain (gluing transformations).

The domain is described as a list of rational cusps, with \infty the cusp at infinity. These are vertices of the domain. The domain is constructed by connecting consecutive cusps with a hyperbolic line (half circle) to create sides. The symbol means that this side is glued to itself by an element of order 2, and there is an additional vertex on the side at the invariant point. The symbol means that there is an additional vertex which is the invariant point of an element of order 3, and the two cusps are connected to it by hyperbolic half-lines.

The generators are described in a table, each generator in a line (c d a b), where (c,d) is the bottom row of the matrix and (a,b) is the top row. Note that c is always a multiple of N. The generators are listed in the same order as the sides of the domain. The list contains pairs of inverse generators since the sides are glued in pairs. 2-torsion generators are inverse to themselves and are listed once. 3-torsion generators are listed without inverse, conforming to the notation of sides.

For &Gamma₀(N) I used the fact that multiplication by -1 is allowed, and all generators are listed with c>0.

The range of N covered will be increased over time. If you have corrections, suggestions or questions write me.

Assaf Wool
email: [email protected]