Rationals and the Modular Group
Roger C. Alperin · American Mathematical Monthly · 1999
The modular group k is the quotient group PSL2(Z) = SL,(Z)/{I we refer to these as reduced strings. We explained this free product structure in terms of the action of the modular group on the irrationals. In this note we describe the action on the rationals; this can be viewed as a way of describing the inverse of the Euclidean algorithm. The group SL2(Z) acts via linear transformations on R2 as column vectors and this gives an action of .A' via linear fractional transformations on the projective line P'(R), the real numbers together with m. We may also view P'(R) as the slopes of non-zero vectors, that is, the equivalence classes of R2 -(:)induced by non-zero scalar multiplication; the equivalence class of the vector e = ( i ) is , ,