In chapter 2 of Three-dimensional Orbifolds and Cone-Manifolds, theorem 2.26 states that
complete geometric orbifolds $Q$ modeled on $(G,X)$, whith $X$ simply connected, are such that the holonomy representation $h:\pi_1^{orb}\to G$ is a isomorphism into a discrete subgroup $\Gamma<G$ which acts properly discontinuously on $X$.
Suppose $Q$ is a $2$-dimensional hyperbolic orbifold, i.e. modeled on $(Isom(\mathbb{H}^2),\mathbb{H}^2)$. I know that $\pi_1^{orb}(Q)$ may contain torsion elements, that hence are elliptic isometries of $\mathbb{H}^2$ and so have fixed points.
But eventually non-trivial elements acting properly discontinuously cannot fix any point by definition, so the question is, what am I missing/getting wrong?
I'm pretty sure that this is a stupid question, but I can't help to find my bug...
It depends on the definition of properly discontinuous action that one uses. In literature are found many non-equivalent definitions (cf. this discussion on properly discontinuous action on mathoverflow).
In particular in this case the confusion arise from the fact that some authors require a properly discontinuous action of a group $G$ on a topological space $X$ to be such that every point $x\in X$ has a neighbourhood $U_x$ such that the only element $g\in G$ such that the property $g.U_x\cap U_x\neq\emptyset$ holds is the identity of $G$, while other authors allow that porperty to hold for finitely many elements.
The linked question discuss pros, cons, and relations between variuous definitions.