Figure 2: The figure on the left hand side displays the action of the modular group
on the complex upper half plane . The two generators of
are and , acting as follows on the coordinate , i.e.,
as an inversion and a translation, respectively. The shaded area indicates the fundamental
domain for the action of on .
The figure on the right hand side displays the action of the “extended modular group”
on . The generators of are obtained by augmenting the generators
of with the generator , acting as on . The additional two
generators of then become: , and their actions on
are . The new generator corresponds to a reflection in the line
, the generator is in turn a reflection in the line , while the generator
is a reflection in the unit circle . The fundamental domain of is
, corresponding to half the fundamental domain of
. The “walls” and correspond, respectively, to the gravity
wall , the symmetry wall and the symmetry wall of Figure 1.
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