probably this question is totally easy and obvious but I am very confused at the moment. So assume we have a matrix $\gamma$ in $SL_2(\mathbb{R})$ acting on the usual upper half-plane $\mathcal{H}$ by Möbius transformation. My question is now, if the map $\gamma: \mathcal{H} \rightarrow \mathcal{H}$ is homotopic to the identity map.
Thanks for granted :)
