 Solution

For the Möbius action on the upper half-plane, a noncentral element is elliptic if it has one fixed point in the interior and none on the boundary, parabolic if it has one boundary fixed point, and hyperbolic if it has two boundary fixed points. Examples areFixed points satisfy a real quadratic. Its discriminant is $(\mathrm{tr}g{)}^2-4$, so the three mutually exclusive cases $|\mathrm{tr}g|<2$, $=2$, and $>2$ give precisely the three types.

If $g^n=I$ and $g\ne \pm I$, its eigenvalues are nonreal reciprocal roots of unity. Thus $|\mathrm{tr}g|<2$ and $g$ is elliptic; no such element is parabolic or hyperbolic.