Hello everyone,

Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of uniform convergence.

Let $h$ be in $\mathcal{A}$. Since $g \geq 2$, $S_g$ is covered by $D^2 \simeq \mathbb{R}^2$, and $h$ can be pulled back on $D^2$ to $\tilde{h}$ in such a way that the projection $(D^2, \tilde{h}) \longrightarrow (S_g , h)$ is a local isometry, and $\pi_1(S_g) \simeq \Gamma \subset \text{Iso}((D^2,\tilde{h}))$ acts on $(D_2, \tilde{h})$ by isometry, so $S_g = D^2 / \Gamma$

My question is, how big is $\Gamma$ in $ \text{Iso}((D^2,\tilde{h}))$ ? For example if one restricts $\mathcal{A}$ to a relevant subset that can be endowed with a good probability measure (to be defined), can one say if $\Gamma = \text{Iso}((D^2,\tilde{h}))$ with probability $1$ ? Or if $[\Gamma :\text{Iso}((D^2,\tilde{h}))] < \infty $? More generally, can one compare $\Gamma$ and $ \text{Iso}((D^2,\tilde{h}))$ for a randomly chosen metric on $S_g$ in any way? I

t's clear for example that in the hyperbolic case, $\Gamma$ is very small in $\textbf{PSl}_2(\mathbb{R})$, but in the general case, one can expect that $\text{Iso}(D^2,\tilde{h})$ is small since the geometry on a random Riemannian manifold has very few symmetries.

Maybe this question is very classic, in that case I would be very gratefull if anyone could give me references on the topic. Thanks for the future answers !