Suppose $f: S \rightarrow S$ is a holomorphic map of a hyperbolic Riemann surface into itself. Suppose $p$ is a fixed point of $f$. I'm trying to understand the modulus of the derivative of $f$ depending on if $f$ decreases Poincaré distances and when $f$ is a local isometry.
In the case where $f$ decreases distances, I lifted $f$ to the universal cover to get $g: \mathbb{D} \rightarrow \mathbb{D}$. Then if we suppose $0 \in f^{-1}(\{p \})$, can we conclude by Pick's Lemma that $|g'(0)|< \frac{1-|g(0)|^2}{1-0^2} \leq 1$ so that $|f'(p)| < 1$? I'm not sure if this is the correct logic and would appreciate any feedback.
In the case where $f$ is a local isometry, I want to show that $|f'(p)| = 1$ but I am running into trouble because if I follow my reasoning from above there is no way to conclude that $g(0) = 0$ here. If this were the case, I would be able to have equality where the first inequality is.