Sl(2,z) matrices with integer coefficients

[JSG31883]

Let SL(2,Z) be the set of 2x2 matrices with integer coefficients.
I know that SL(2,Z) is generated by S and T, where
S= (0 -1
1 0)
and T= (1 1
0 1).

But how can I show that everyone element of G (the group generated by S and T) is in SL(2,Z)?

Also, let FcH (upper half-plane) be defined as F= {z in C: abs(z)>1, abs(Re(z)<1/2)}.
How can I draw a picture of F? Which linear fractional transformations correspond to S and T (as given above)?

 

[matt grime]

Am i reading riht? tw group generrated by S and T is the set of all products of S, T and their inverses in some order some fininte number of times. the entries are boviously all integers and the derteminants all1 so of course it is in SL(2,Z)

and the second part... well, as alwayas, draw on the regiosn where |z|=1, and |Re(z)|=1/2 and work out which region corresponds to the inequialities

 

[thepatientmental]

To show that every element of G is in SL(2,Z), we can use the fact that the determinant of any element in SL(2,Z) must be equal to 1. Since both S and T have determinant equal to 1, any combination of these two matrices will also have determinant equal to 1. This means that all elements of G are in SL(2,Z).

To draw a picture of F, we can use the fact that F is the set of complex numbers with absolute value greater than 1 and real part less than 1/2. This can be visualized as a region in the complex plane that is outside of the unit circle and to the left of the line Re(z)=1/2. This region can be shaded or outlined to represent F.

To determine the linear fractional transformations corresponding to S and T, we can use the fact that these transformations are given by the formula z'=(az+b)/(cz+d), where a,b,c,d are the entries of the 2x2 matrix. For S, we have a=0, b=-1, c=1, and d=0. Plugging these values into the formula gives us z'=-1/z, which is the reflection of z about the imaginary axis. For T, we have a=1, b=1, c=0, and d=1. Plugging these values into the formula gives us z'=z+1, which is a translation to the right by 1 unit.