摘要
In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of amalgamation of the components of F is also given. Some elements a＋b√3/c of Q（i, √3） and their conjugates a-b√3/c a c over Q（i） have different signs in the orbits of the biquadratic field Q（i, √3） when acted upon by F. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Fa, and they form a closed path which is the only closed path in the orbit Гa. We also devise a procedure to obtain ambiguous numbers of the form a-b√3/c, where b is a positive integer.
In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of amalgamation of the components of F is also given. Some elements a＋b√3/c of Q（i, √3） and their conjugates a-b√3/c a c over Q（i） have different signs in the orbits of the biquadratic field Q（i, √3） when acted upon by F. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Fa, and they form a closed path which is the only closed path in the orbit Гa. We also devise a procedure to obtain ambiguous numbers of the form a-b√3/c, where b is a positive integer.
