Write down the Riemannian metric for the upper half-plane model $\mathbf{H}$ of the hyperbolic plane. Describe, without proof, the group of isometries of $\mathbf{H}$ and the hyperbolic lines (i.e. the geodesics) on $\mathbf{H}$.

Show that for any two hyperbolic lines $\ell_{1}, \ell_{2}$, there is an isometry of $\mathbf{H}$ which maps $\ell_{1}$ onto $\ell_{2}$.

Suppose that $g$ is a composition of two reflections in hyperbolic lines which are ultraparallel (i.e. do not meet either in the hyperbolic plane or at its boundary). Show that $g$ cannot be an element of finite order in the group of isometries of $\mathbf{H}$.

[Existence of a common perpendicular to two ultraparallel hyperbolic lines may be assumed. You might like to choose carefully which hyperbolic line to consider as a common perpendicular.]