= Solution
Take the <parabolic element of PSL2(R)>
$$
A=\begin{pmatrix}1&1\\0&1\end{pmatrix},
\qquad z\longmapsto z+1.
$$
It fixes no point in $\mathbb H$. A <Geodesic in the Poincare half-plane model> is determined by its unordered pair of ideal endpoints: either two real numbers or one real number together with $\infty$. Translation adds one to every finite endpoint, so no such pair is invariant. Therefore \b[this $A$ preserves neither a point nor a hyperbolic line in $\mathbb H$.]
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