Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following:
(1) $\rho:\pi_1(S)\hookrightarrow Isom^+(\mathbb{H}^2)$,
(2) $\forall \gamma\in \pi_1(S)$, $\gamma\ne Id$, then $\rho(\gamma)$ is hyperbolique.
Thanks in advance.