Let $M_{g}$ be the compact Riemann surface with $g\geq 2$. Is there an infinit group $G$ with an equivariant action on the pair $(TM_{g}, M_{g})$ such that the action on the fibers preserves the inner product, moreover there is no a $G$-fixed point on the base space. Recall that a $G$-fixed point is a point $p$ with $g.p=p$ for all $g\in G$.
This question is a remedy of my previous post bellow
