Evolution of ambiguous numbers under the actions of a Bianchi group

Abstract

In this paper we study some combinatorial properties of biquadratic irrational number field ${\mathbb{Q}}( {i,\sqrt{3} } ) $, under the action of a Bianchi group ${\Gamma _3} = PSL({2,{O_3}} ) $. In this experiment it is revealed that a special class of elements exists; that is, for an element $\xi $ its conjugate $\bar{\xi } $ has different signs in the closed path (orbits) for the action of ${\Gamma _3} $ over ${\mathbb{Q}}( {i,\sqrt{3} } ) $, known as ambiguous numbers. It is also proved that the orbit $\Gamma \xi $ defined on a finite number of ambiguous numbers succeeding a unique closed path.

Keywords

• group action
• bianchi group
• biquadratic irrational field
• orbits