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

中文翻译：

---

在比安奇集团的作用下模棱两可数字的演化

本文研究了双二次无理数场的一些组合性质 $\mathbb{问}（\mathrm{一世}，\sqrt{3}）$在比安奇集团的推动下 ${\mathrm{\Gamma }}_3=P\mathrm{小号}\mathrm{大号}（2，{Ø}_3）$。在该实验中，发现存在一类特殊的元素。也就是说，对于一个元素$\xi$ 其共轭 $\overline{\xi }$ 在闭合路径（轨道）中具有不同的符号 ${\mathrm{\Gamma }}_3$ 过度 $\mathbb{问}（\mathrm{一世}，\sqrt{3}）$，称为歧义数。也证明了轨道$\mathrm{\Gamma }\xi$ 在唯一闭合路径之后的有限数量的模糊数上定义。