Sl(2,z) matrices with integer coefficients

  • Thread starter JSG31883
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  • #1
JSG31883
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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)?
 

Answers and Replies

  • #2
matt grime
Science Advisor
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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
 

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