I'm referencing material from: Dynamics of geodesic and horocyclic flows and Hyperbolic coordinates.
The upper half plane can be given a hyperbolic structure and since there is a bijection to the first quadrant using hyperbolic coordinates, horocycles in the upper half plane can be corresponded to hyperbola in the first quadrant. So from what I understand the horocyclic flow transforms points and moves them along horocycles, which I will correspond to hyperbolas in the first quadrant.
Why then is the one parameter matrix flow of the horocyclic flow given by:
$$\bigg\{\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix},s\in\Bbb R\bigg\}?$$
It would seem to me that the horocylic flow should be given by:
$$\bigg\{\begin{pmatrix} e^{t/2} & 0 \\ 0 & e^{-t/2} \end{pmatrix},t\in\Bbb R\bigg\}.$$
Because this matrix "moves" points along a hyperbola.
But the author in the first link states this second matrix is the one parameter matrix flow of the geodesic flow.