Figure 2

Figure 2: The figure on the left hand side displays the action of the modular group P SL (2,ℤ ) on the complex upper half plane ℍ = {z ∈ ℂ |ℑz > 0}. The two generators of P SL (2,ℤ ) are S and T, acting as follows on the coordinate z ∈ ℍ : S(z) = − 1∕z; T(z) = z + 1, i.e., as an inversion and a translation, respectively. The shaded area indicates the fundamental domain 𝒟P SL (2,ℤ) = {z ∈ ℍ | − 1∕2 ≤ ℜz ≤ 1∕2; |z| ≥ 1} for the action of PSL (2,ℤ ) on ℍ. The figure on the right hand side displays the action of the “extended modular group” P GL (2,ℤ ) on ℍ. The generators of P GL (2,ℤ ) are obtained by augmenting the generators of P SL (2,ℤ ) with the generator s1, acting as s1(z) = − ¯z on ℍ. The additional two generators of P GL (2,ℤ) then become: s2 ≡ s1 ∘ T ; s3 ≡ s1 ∘ S, and their actions on ℍ are s2(z) = 1 − ¯z; s3(z ) = 1 ∕¯z. The new generator s1 corresponds to a reflection in the line ℜz = 0, the generator s2 is in turn a reflection in the line ℜz = 1∕2, while the generator s 3 is a reflection in the unit circle |z| = 1. The fundamental domain of P GL (2,ℤ ) is 𝒟P GL (2,ℤ) = {z ∈ ℍ |0 ≤ ℜz ≤ 1∕2; |z| ≥ 1}, corresponding to half the fundamental domain of P SL (2,ℤ ). The “walls” ℜz = 0,ℜz = 1∕2 and |z| = 1 correspond, respectively, to the gravity wall α1(β ) = 0, the symmetry wall α2(β ) = 0 and the symmetry wall α3(β ) = 0 of Figure 1.