Evolution of ambiguous numbers under the actions of a Bianchi group. Issue 1 (1st January 2020) - Yousaf, Awais; Alolaiyan, Hanan; Razaq, Abdul; Younis, Muhammad
Abstract : 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.