Affiliation:
1. Department of Mathematics, Necatibey Faculty of Education, Balıkesir University , 10100 Balıkesir , Türkiye
Abstract
Abstract
There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group
G
3
,
3
{G}_{3,3}
generated by the linear fractional transformations
a
=
1
−
1
∕
z
a=1-1/z
and
b
=
z
+
2
b=z+2
. This group is the unique subgroup of the modular group
PSL
(
2
,
Z
)
{\rm{PSL}}(2,{\mathbb{Z}})
with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number
p
∕
q
p/q
, to obtain an element in
G
3
,
3
{G}_{3,3}
with cusp point
p
∕
q
p/q
. As a result, we say that the action of
G
3
,
3
{G}_{3,3}
on rational numbers is transitive.
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