 Solution

Take the parabolic element of PSL2(R)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 this $A$ preserves neither a point nor a hyperbolic line in $\mathbb{H}$.