Advanced book on Mathematics Olympiad

272 5 Number Theory

has determinant 1. The matrices with this property form thespecial linear groupSL( 2 ,Z).

This group is generated by the matrices

S=

(

0 − 1

10

)

and T=

(

11

01

)

.

Explicitly,

(

ay

bx

)

=STa^1 STa^2 S···STanS,

since matrix multiplication mimics the (backward) calculation of the continued fraction.
We thus have a method of expressing the elements of SL( 2 ,Z)in terms of generators.
The special linear group SL( 2 ,Z)arises in non-Euclidean geometry. It acts on the
upper half-plane, on which Poincaré modeled the “plane’’ of Lobachevskian geometry.
The “lines’’ of this “plane’’ are the semicircles and half-lines orthogonal to the real axis.
A matrix

A

(

ab

cd

)

acts on the Lobachevski plane by

z→

az+b

cz+d

,ad−bc= 1.

All these transformations form a group of isometries of the Lobachevski plane. Note
thatAand−Ainduce the same transformations; thus this group of isometries of
the Lobachevski plane, also called the modular group, is isomorphic to PSL( 2 ,Z)=
SL( 2 ,Z)/{−I 2 ,I 2 }. The matricesSandTbecome the inversion with respect to the unit
circlez→−^1 zand the translationz→z+1.
We stop here with the discussion and list some problems.

794.Write the matrix

(

12 5

73

)

as the product of several copies of the matrices

(

01

10

)

and

(

11

01

)

.

(No, there is no typo in the matrix on the left.)