双曲几何学 Hyperbolic Geometry MATH32051

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If $m$ is either elliptic or loxodromic, we may write $n(z)=p \circ m \circ p^{-1}(z)=\alpha^{2} z$, where $\alpha^{2} \in \mathbb{C}-{0,1}$. When we normalize so that the determinant of $n$ (or equivalently of $m$ ) is 1 , we need to write
$$n(z)=\frac{\alpha z}{\alpha^{-1}},$$
and so
$$\tau(n)=\tau\left(p \circ m \circ p^{-1}\right)=\tau(m)=\left(\alpha+\alpha^{-1}\right)^{2}$$

In the case in which $m$ is elliptic, so that $|\alpha|=1$, write $\alpha=e^{i \theta}$ for some $\theta$ in $(0, \pi)$. Calculating, we see that
$$\tau(m)=\left(\alpha+\alpha^{-1}\right)^{2}=\left(e^{i \theta}+e^{-i \theta}\right)^{2}=4 \cos ^{2}(\theta)$$
In particular, we have that $\tau(m)$ is real and lies in the interval $[0,4)$.
In the case in which $m$ is loxodromic, so that $|\alpha| \neq 1$, we write $\alpha=\rho e^{i \theta}$ for some $\rho>0, \rho \neq 1$, and some $\theta$ in $[0, \pi)$. Calculating, we see that
$$\alpha+\alpha^{-1}=\rho e^{i \theta}+\rho^{-1} e^{-i \theta}$$

MATH32051 COURSE NOTES ：

Note that in the latter case, because $C(\overline{\mathbb{R}})=\overline{\mathbb{R}}$, we may instead consider the composition
$$m \circ C(z)=m(\bar{z})=\frac{a z+b}{c z+d},$$
and so reduce ourselves to considering just the former case in which
$$m(z)=\frac{a z+b}{c z+d},$$
where $a, b, c, d \in \mathbb{C}$ and $a d-b c=1$.