# Connected group action on hyperbolic surface

Is it possible to have a faithful action of a connected nontrivial Lie group on a hyperbolic surface $$\Sigma_g$$?

Some background:

It is impossible to have a transitive action of a connected nontrivial Lie group on a hyperbolic surfaces $$\Sigma_g$$.

It is impossible to have a faithful action of a nontrivial connected Lie group by isometries or by conformal transformations on $$\Sigma_g$$.

I think any action by a compact group is in some sense an action by isometries so every compact connected group must act trivially on $$\Sigma_g$$.

This is perhaps related to the fact that $$\Sigma_g$$ are spin manifolds and a theorem of Atiyah and Hirzebruch that for certain spin manifolds of dimension $$4k$$ (obviously doesn't directly apply since dimension here is $$2$$ but maybe it's a similar idea) every action by a compact connected group is trivial. See https://mathoverflow.net/a/89358/387190

• What does nontrivial mean? The group $\mathbb{R}$ acts on the closed unit disk in the complex plane by the formula $\phi_t(z) = e^{i(1-|z|)t}z$. Since this action is the identity on the boundary of the disk, this can be done on any closed disk contained in any surface. Commented Apr 20, 2022 at 21:57
• @JimBelk that's a good point. Do you want to just post that as an answer? Commented Sep 29, 2022 at 16:53

Yes. The group $$\mathbb{R}$$ acts on the closed unit disk in the complex plane by the formula $$\phi_t(z) = e^{(1-|z|)t}z$$. Since this action is the identity on the boundary of the disk, this can be done on any closed disk contained on any surface.