  Let $X$ be a hyperbolic Riemann surface. We study a convergent Wick-type star
product $\star_X$ on $X$ which is induced by the canonical convergent star
product $\star_{\mathbb{D}}$ on the unit disk $\mathbb{D}$ via Uniformization
Theory. While by construction, the resulting Fr\'echet algebras
$(\mathcal{A}(X),\star_X)$ are strongly isomorphic for conformally equivalent
Riemann surfaces, our work exhibits additional severe topological obstructions.
In particular, we show that the Fr\'echet algebra $(\mathcal{A}(X),\star_X)$
degenerates if and only if the connectivity of $X$ is at least $3$, and
$(\mathcal{A}(X),\star_X)$ is noncommutative if and only if $X$ is simply
connected. We also explicitly determine the algebra $\mathcal{A}_X$ and the
star product $\star_X$ for the intermediate case of doubly connected Riemann
surfaces $X$. As a perhaps surprinsing result, we deduce that two such
Fr\'echet algebras are strongly isomorphic if and only if either both Riemann
surfaces are conformally equivalent to an (not neccesarily the same) annulus or
both are conformally equivalent to a punctured disk.
