Evolution of ambiguous numbers under the actions of a Bianchi group.

In this paper we study some combinatorial properties of biquadratic irrational number field Q (i , 3) , under the action of a Bianchi group Γ 3 = P S L (2 , O 3 ). In this experiment it is revealed that a special class of elements exists; that is, for an element ξ its conjugate ξ ¯ has different signs in the closed path (orbits) for the action of Γ 3 over Q (i , 3) , known as ambiguous numbers. It is also proved that the orbit Γ ξ defined on a finite number of ambiguous numbers succeeding a unique closed path. [ABSTRACT FROM AUTHOR]