Hecke Operators and the Fundamental Domain forSL ( 3 , Z )

A fundamental domain D for T2 is a connected closed subset D c H behaving like the quotient space T2 \ H, at least up to boundary identifications. Thus, for every z g H, there is a y G T2 such that yz g D. Moreover, if z and w lie in the interior of D and z = yw for y G T2, then y = ±1, where / is the identity matrix. It is easily seen (cf. Terras [17]) that the region (1.1) F2= (zg//| -i< Rez<±, |z|> 1} is a fundamental domain for SL(2, Z). The usual method of moving z G H to F2 is called the "highest-point method", i.e., you choose y g T2 to maximize Im(yz). The process of moving z to yz g D is called a reduction algorithm. It can be done by a sequence of flips by H°, I) and translations T-(l I). „ez.