3.II.14B

Describe the hyperbolic lines in the upper half-plane model $H$ of the hyperbolic plane. The group $G=\mathrm{SL}(2, \mathbb{R}) /\{\pm I\}$ acts on $H$ via Möbius transformations, which you may assume are isometries of $H$. Show that $G$ acts transitively on the hyperbolic lines. Find explicit formulae for the reflection in the hyperbolic line $L$ in the cases (i) $L$ is a vertical line $x=a$, and (ii) $L$ is the unit semi-circle with centre the origin. Verify that the composite $T$ of a reflection of type (ii) followed afterwards by one of type (i) is given by $T(z)=-z^{-1}+2 a$.

Suppose now that $L_{1}$ and $L_{2}$ are distinct hyperbolic lines in the hyperbolic plane, with $R_{1}, R_{2}$ denoting the corresponding reflections. By considering different models of the hyperbolic plane, or otherwise, show that

(a) $R_{1} R_{2}$ has infinite order if $L_{1}$ and $L_{2}$ are parallel or ultraparallel, and

(b) $R_{1} R_{2}$ has finite order if and only if $L_{1}$ and $L_{2}$ meet at an angle which is a rational multiple of $\pi$.

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