Number Theory: An Introduction to Mathematics

5 The Modular Group 203

SinceS−^1 =Sand(Tn)−^1 =T−n, it follows that

g=Tm^1 S···TmkSTmorg=Tm^1 S···TmkTm. 

The proof of Proposition 12 may be regarded as an analogue of the continued
fraction algorithm, since

Tm^1 S···TmkSTmz=m 1 −

1

m 2 −

..

.

−

1

mk−

1

m+z

.

ObviouslyΓ is also generated bySandR := ST. The transformationRhas
order 3, since

R(z)=− 1 /(z+ 1 ), R^2 (z)=−(z+ 1 )/z, R^3 (z)=z.

We are going to show that all other relations between the generatorsSandRare
consequences of the relationsS^2 =R^3 =I,sothatΓis thefree productof a cyclic
group of order 2 and a cyclic group of order 3.
Partition the upper half-planeHby putting

A={z∈H :Rz< 0 }, B={z∈H :Rz≥ 0 }.

It is easily verified that

SA⊂B, RB⊂A, R^2 B⊂A

(where the inclusions are strict). Ifg′=SRε^1 SRε^2 ···SRεnfor somen≥1, where
εj∈{ 1 , 2 }, it follows thatg′B⊂Bandg′SA⊂B. Similarly, ifg′′=Rε^1 S···Rεn,
theng′′B⊂Aandg′′SA⊂A. By taking account of the relationsS^2 =R^3 =I,every
g∈Γcan be written in one of the forms

I,S,g′,g′′,g′S,g′′S.

But, by what has just been said, no element except the first is the identity transforma-
tion.
The modular group isdiscrete, since there exists a neighbourhood of the identity
transformation which contains no other element ofΓ.

Proposition 13The open set

F={z∈H :− 1 / 2 <Rz< 1 / 2 ,|z|> 1 }

(see Figure 1) is a fundamental domain for the modular groupΓ, i.e. distinct points
of F are not equivalent and each point ofH is equivalent to some point of F or its
boundary∂F.