I am trying to prove the following.
Let $H\subset\mathbb C$ be the upper half plane (equipped with the hyperbolic metric $\frac{1}{y^2}dzd\bar z$), ${\rm PSL}(2,\mathbb R)$ the transformation group of $H$, and $\Gamma\subset{\rm PSL}(2,\mathbb R)$ a subgroup that acts properly discontinuous and is free from fixed points. Then
(1) for each $z_0\in H$, it has a neighborhood $U$ such that $g(U)\cap U=\varnothing$ for all $g\neq{\rm id}$ in $\Gamma$, and the restriction of the canonical projection to $U$: $$\pi|_U:U\to\pi(U)$$ is a homeomorphism, and thus is a local chart for $H/\Gamma$. And all charts of this type constitute an atlas that makes $H/\Gamma$ a Riemann surface;
(2) this procedure also equips $H/\Gamma$ with a hyperbolic metric.
Questions:
(1a) How does $$\forall z_0\in H,\exists\text{ nbhd }U\text{ of }z_0,\text{ s.t. }\forall g\in\Gamma, g\neq{\rm id},\ g(U)\cap U=\varnothing$$ follow from $\Gamma$ acting properly discontinuous and being free from fixed points? The rest I can understand just fine. (Actually I already have a proof below, but the book makes it look so obvious that I am looking for a more concise and elegant one if there is one).
(1b) Why is $gU\cap U=\varnothing$ necessary for $\pi|_U$ to be a homeomorphism?
(2) How does this procedure give $H/\Gamma$ a hyperbolic metric? More specifically, why is it well-defined?
Attempts:
(1a) With $\Gamma$ being properly discontinuous, for any $z_0\in H$ we can find a neighborhood $U_0$ such that $\{g\in\Gamma:gU_0\cap U_0\neq\varnothing\}$ is a finite set. If it contains only ${\rm id}$ then we are done. If not, let the elements be $$g_1={\rm id},\ g_2,\cdots,\ g_n$$ Now by the Hausdorff property of $H$ and the fact that $\Gamma$ is free from fixed points we find nonintersecting neighborhoods $$U_1,\ \cdots,\ U_n$$ of $$z_0,\ g_2z_0,\ \cdots,\ g_nz_0$$ Finally let $U=U_0\cap(\bigcap_{k=1}^ng_k^{-1}U_k)$.
Is this correct? Is there a more trivial proof of this part?
(1b) I think this is to ensure $\pi|_U:U\to\pi(U)$ is injective.
(2) Is it enough to say all elements in $\Gamma$ are isometries of $H$ for the hyperbolic metric $\frac{1}{y^2}dzd\bar z$?